case study for quadratic equation

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CBSE Class 10 Maths Case Study Questions for Chapter 4 Quadratic Equations (Published by CBSE)

Cbse class 10 maths case study questions for chapter 4 - quadratic equations are released by the board. solve all these questions to perform well in your cbse class 10 maths exam 2021-22..

Check here the case study questions for CBSE Class 10 Maths Chapter 4 - Quadratic Equations. The board has published these questions to help class 10 students to understand the new format of questions. All the questions are provided with answers. Students must practice all the case study questions to prepare well for their Maths exam 2021-2022.

Case Study Questions for Class 10 Maths Chapter 4 - Quadratic Equations

CASE STUDY 1:

Raj and Ajay are very close friends. Both the families decide to go to Ranikhet by their own cars. Raj’s car travels at a speed of x km/h while Ajay’s car travels 5 km/h faster than Raj’s car. Raj took 4 hours more than Ajay to complete the journey of 400 km.

1. What will be the distance covered by Ajay’s car in two hours?

 a) 2(x + 5)km

b) (x – 5)km

c) 2(x + 10)km

d) (2x + 5)km

Answer: a) 2(x + 5)km

2. Which of the following quadratic equation describe the speed of Raj’s car?

a) x 2 – 5x – 500 = 0

b) x 2 + 4x – 400 = 0

c) x 2 + 5x – 500 = 0

d) x 2 – 4x + 400 = 0

Answer: c) x 2 + 5x – 500 = 0

3. What is the speed of Raj’s car?

a) 20 km/hour

b) 15 km/hour

c) 25 km/hour

d) 10 km/hour

Answer: a) 20 km/hour

4. How much time took Ajay to travel 400 km?

Answer: d) 16 hour

CASE STUDY 2:

The speed of a motor boat is 20 km/hr. For covering the distance of 15 km the boat took 1 hour more for upstream than downstream.

1. Let speed of the stream be x km/hr. then speed of the motorboat in upstream will be

a) 20 km/hr

b) (20 + x) km/hr

c) (20 – x) km/hr

Answer: c) (20 – x)km/hr

2. What is the relation between speed ,distance and time?

a) speed = (distance )/time

b) distance = (speed )/time

c) time = speed x distance

d) speed = distance x time

Answer: b) distance = (speed )/time

3. Which is the correct quadratic equation for the speed of the current?

a) x 2 + 30x − 200 = 0

b) x 2 + 20x − 400 = 0

c) x 2 + 30x − 400 = 0

d) x 2 − 20x − 400 = 0

Answer: c) x 2 + 30x − 400 = 0

4. What is the speed of current ?

b) 10 km/hour

c) 15 km/hour

d) 25 km/hour

Answer: b) 10 km/hour

5. How much time boat took in downstream?

a) 90 minute

b) 15 minute

c) 30 minute

d) 45 minute

Answer: d) 45 minute

Also Check:

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CBSE 10th Standard Maths Subject Quadratic Equations Case Study Questions 2021

By QB365 on 21 May, 2021

QB365 Provides the updated CASE Study Questions for Class 10 Maths, and also provide the detail solution for each and every case study questions . Case study questions are latest updated question pattern from NCERT, QB365 will helps to get  more marks in Exams

QB365 - Question Bank Software

10th Standard CBSE

Final Semester - June 2015

Case Study Questions

A quadratic equation can be defined as an equation of degree 2. This means that the highest exponent of the polynomial in it is 2. The standard form of a quadratic equation is ax 2 + bx + c = 0, where a, b, and c are real numbers and  \(a \neq 0\)   Every quadratic equation has two roots depending on the nature of its discriminant, D = b2 - 4ac.Based on the above information, answer the following questions. (i) Which of the following quadratic equation have no real roots?

(ii) Which of the following quadratic equation have rational roots?

(iii) Which of the following quadratic equation have irrational roots?

(iv) Which of the following quadratic equations have equal roots?

(v) Which of the following quadratic equations has two distinct real roots?

In our daily life we use quadratic formula as for calculating areas, determining a product's profit or formulating the speed of an object and many more. Based on the above information, answer the following questions. (i) If the roots of the quadratic equation are 2, -3, then its equation is

(ii) If one root of the quadratic equation 2x 2 + kx + 1 = 0 is -1/2, then k =

(iii) Which of the following quadratic equations, has equal and opposite roots?

(iv) Which of the following quadratic equations can be represented as (x - 2) 2 + 19 = 0?

(v) If one root of a qua drraattiic equation is  \(\frac{1+\sqrt{5}}{7}\) , then I.ts other root is

Quadratic equations started around 3000 B.C. with the Babylonians. They were one of the world's first civilisation, and came up with some great ideas like agriculture, irrigation and writing. There were many reasons why Babylonians needed to solve quadratic equations. For example to know what amount of crop you can grow on the square field; Based on the above information, represent the following questions in the form of quadratic equation. (i) The sum of squares of two consecutive integers is 650.

(ii) The sum of two numbers is 15 and the sum of their reciprocals is 3/10.

(iii) Two numbers differ by 3 and their product is 504.

(iv) A natural number whose square diminished by 84 is thrice of 8 more of given number.

(v) A natural number when increased by 12, equals 160 times its reciprocal.

Amit is preparing for his upcoming semester exam. For this, he has to practice the chapter of Quadratic Equations. So he started with factorization method. Let two linear factors of  \(a x^{2}+b x+c \text { be }(p x+q) \text { and }(r x+s)\) \(\therefore a x^{2}+b x+c=(p x+q)(r x+s)=p r x^{2}+(p s+q r) x+q s .\) Now, factorize each of the following quadratic equations and find the roots. (i) 6x 2 + x - 2 = 0

(ii) 2x 2 -+ x - 300 = 0

(iii) x 2 -  8x + 16 = 0

(iv) 6x 2 -  13x + 5 = 0

(v) 100x 2 - 20x + 1 = 0

If p(x) is a quadratic polynomial i.e., p(x) = ax 2 - + bx + c, \(a \neq 0\) , then p(x) = 0 is called a quadratic equation. Now, answer the following questions. (i) Which of the following is correct about the quadratic equation ax 2 - + bx + c = 0 ?

(ii) The degree of a quadratic equation is

(iii) Which of the following is a quadratic equation?

(iv) Which of the following is incorrect about the quadratic equation ax 2 - + bx + c = 0 ?

(v) Which of the following is not a method of finding solutions of the given quadratic equation?

*****************************************

Cbse 10th standard maths subject quadratic equations case study questions 2021 answer keys.

(i) (a): To have no real roots, discriminant (D = b 2 - 4ac) should be < 0. (a) D = 7 2 - 4(-4)(-4) = 49 - 64 = -15 < 0 (b) D=7 2 -4(-4)(-2)=49-32=17>0 (c) D = 5 2 - 4(-2)(-2) = 25 - 16 = 9 > 0 (d) D = 6 2 - 4(3)(2) = 36 - 24 = 12> 0 (ii) (b): To have rational roots, discriminant (D = b 2 - 4ac) should be> 0 and also a perfect square (a) D = 1 2 - 4(1)( -1) = 1 + 4 = 5, which is not a perfect square. (b) D = (-5) 2 - 4(1)(6) = 25 - 24 = I, which is a perfect square. (c) D = (-3) 2  - 4(4)(-2) = 9 + 32 = 41, which is not a perfect square. (d) D = (-1) 2 - 4(6)(11) = 1 - 264 = -263, which is not a perfect square. (iii) (c) : To have irrational roots, discriminant (D = b 2 - 4ac) should be > 0 but not a perfect square. (a) D = 2 2 - 4(3)(2) = 4 - 24 = -20 < 0 (b) D = (-7) 2 - 4(4)(3) = 49 - 48 = 1 > 0 and also a perfect square. (c) D = (-3) 2 - 4(6)(-5) = 9 + 120 = 129> 0 and not a perfect square. (d) D = 3 2 - 4(2)(-2) = 9 + 16 = 25 > 0 and also a perfect square. (iv) (d): To have equal roots, discriminant (D = b 2 - 4ac) should be = 0. (a) D=(-3) 2 -4(1)(4)=9-16=-7<0 (b) D = (-2) 2 - 4(2)(1) = 4 - 8 = -4 < 0 (c) D = (-10) 2 - 4(5)(1) = 100 - 20 = 80 > 0 (d) D = 6 2 - 4(9)(1) = 36 - 36 = 0 (v) (a): To have two distinct real roots, discriminant (D = b 2 - 4ac) should be > 0. (a) D = 3 2 - 4(1)(1) = 9 - 4 = 5 > 0 (b) D = 3 2 - 4(-1)( -3) = 9 - 12 = -3 < 0 (c) D=8 2 - 4(4)(4) = 64-64 = 0 (d) D = 6 2 - 4(3)(4) = 36 - 48 = -12 < 0

(i) (b): Roots of the quadratic equation are 2 and -3. \(\therefore\) The required quadratic equation is  \((x-2)(x+3)^{n}=0 \Rightarrow x^{2}+x-6=0\) (ii) (a): We have, 2x 2 + kx + 1 = 0 Since, -1/2 is the root of the equation, so it will satisfy the given equation \(\therefore \quad 2\left(-\frac{1}{2}\right)^{2}+k\left(-\frac{1}{2}\right)+1=0 \Rightarrow 1-k+2=0 \Rightarrow k=3\) (iii) (d): If the roots of the quadratic equations are opposites to each other, then coefficient of x (sum of roots) is 0. So, both (a) and (b) have the coefficient of x = 0. (iv) (c): The given equation is (x - 2) 2 + 19 = 0 \(\Rightarrow x^{2}-4 x+4+19=0 \Rightarrow x^{2}-4 x+23=0\) (v) (b): If one root of a quadratic equation is irrational, then its other root is also irrational and also its conjugate i.e., if one root is p +. \(\sqrt(q)\)  then its other root is p -. \(\sqrt(q)\) .

(i) (b): Let two consecutive integers be x, x + 1. Given, x 2 + (x + 1) 2 = 650 \(\begin{array}{l} \Rightarrow 2 x^{2}+2 x+1-650=0 \\ \Rightarrow 2 x^{2}+2 x-649=0 \end{array}\) (ii) (c): Let the two numbers be x and 15 - x. Given,  \(\frac{1}{x}+\frac{1}{15-x}=\frac{3}{10}\) \(\begin{array}{l} \Rightarrow 10(15-x+x)=3 x(15-x) \\ \Rightarrow 50=15 x-x^{2} \Rightarrow x^{2}-15 x+50=0 \end{array}\) (iii) (d): Let the numbers be x and x + 3. Given, x(x + 3) = 504 \(\Rightarrow\) x 2 + 3x - 504 = 0 (iv) (c): Let the number be x. According to question, x 2 - 84 = 3(x + 8) \(\Rightarrow x^{2}-84=3 x+24 \Rightarrow x^{2}-3 x-108=0\) (v) (d): Let the number be x. According to question, x + 12 =  \(\frac {160}{x}\) \(\Rightarrow x^{2}+12 x-160=0\)

(i) (b): We have  \(6 x^{2}+x-2=0\) \(\Rightarrow \quad 6 x^{2}-3 x+4 x-2=0 \) \(\Rightarrow \quad(3 x+2)(2 x-1)=0 \) \(\Rightarrow \quad x=\frac{1}{2}, \frac{-2}{3}\) (ii) (c):  \(2 x^{2}+x-300=0\) \(\Rightarrow \quad 2 x^{2}-24 x+25 x-300=0 \) \(\Rightarrow \quad(x-12)(2 x+25)=0 \) \(\Rightarrow \quad x=12, \frac{-25}{2}\) (iii) (d):   \(x^{2}-8 x+16=0\) \(\Rightarrow(x-4)^{2}=0 \Rightarrow(x-4)(x-4)=0 \Rightarrow x=4,4\) (iv) (d):   \(6 x^{2}-13 x+5=0\) \(\Rightarrow \quad 6 x^{2}-3 x-10 x+5=0 \) \(\Rightarrow \quad(2 x-1)(3 x-5)=0 \) \(\Rightarrow \quad x=\frac{1}{2}, \frac{5}{3}\) (v) (a):  \(100 x^{2}-20 x+1=0\) \(\Rightarrow(10 x-1)^{2}=0 \Rightarrow x=\frac{1}{10}, \frac{1}{10}\)  

(i) (d) (ii) (b) (iii) (a):  x(x + 3) + 7 = 5x - 11 \(\Rightarrow x^{2}+3 x+7=5 x-11\) \(\Rightarrow x^{2}-2 x+18=0 \)  is a quadratic equation. \((b) (x-1)^{2}-9=(x-4)(x+3)\) \(\Rightarrow x^{2}-2 x-8=x^{2}-x-12\) \(\Rightarrow x-4=0\)   is not a quadratic equation. \((c) x^{2}(2 x+1)-4=5 x^{2}-10\) \(\Rightarrow 2 x^{3}+x^{2}-4=5 x^{2}-10\) \(\Rightarrow 2 x^{3}-4 x^{2}+6=0\)   is not a quadratic equation. \((d) x(x-1)(x+7)=x(6 x-9)\) \(\Rightarrow x^{3}+6 x^{2}-7 x=6 x^{2}-9 x\) \(\Rightarrow x^{3}+2 x=0\)    is not a quadratic equation. (iv) (d) (v) (d)

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Class 10 Maths: Case Study Questions of Chapter 4 Quadratic Equations PDF

Case study Questions on the Class 10 Mathematics Chapter 4  are very important to solve for your exam. Class 10 Maths Chapter 4 Case Study Questions have been prepared for the latest exam pattern. You can check your knowledge by solving case study-based questions for Class 10 Maths Chapter 4 Quadratic Equations

In CBSE Class 10 Maths Paper, Students will have to answer some questions based on  Assertion and Reason . There will be a few questions based on case studies and passage-based as well. In that, a paragraph will be given, and then the MCQ questions based on it will be asked.

Quadratic Equations Case Study Questions With answers

Here, we have provided case-based/passage-based questions for Class 10 Maths  Chapter 4 Quadratic Equations

Case Study/Passage Based Questions

1)Formation of Quadratic Equation

Quadratic equations started around 3000 B.C. with the Babylonians. They were one of the world’s first civilizations and came up with some great ideas like agriculture, irrigation, and writing. There were many reasons why Babylonians needed to solve quadratic equations. For example to know what amount of crop you can grow on the square field. Now represent the following situations in the form of a quadratic equation.

The sum of squares of two consecutive integers is 650. (a) x 2 + 2x – 650 = 0 (b) 2x 2 +2x – 649 = 0 (c) x 2 – 2x – 650 = 0 (d) 2x 2 + 6x – 550 = 0

Answer: (b) 2×2 +2x – 649 = 0

The sum of two numbers is 15 and the sum of their reciprocals is 3/10. (a) x 2 + 10x – 150 = 0 (b) 15x 2 – x + 150 = 0 (c) x 2 – 15x + 50 = 0 (d) 3x 2 – 10x + 15 = 0

Answer: (c) x2 – 15x + 50 = 0

Two numbers differ by 3 and their product is 504. (a) 3x 2 – 504 = 0 (b) x 2 – 504x + 3 = 0 (c) 504x 2 +3 = x (d) x 2 + 3x – 504 = 0

Answer: (d) x2 + 3x – 504 = 0

A natural number whose square diminished by 84 is thrice of 8 more of a given number. (a) x 2 + 8x – 84 = 0 (b) 3x 2 – 84x + 3 = 0 (c) x 2 – 3x – 108 = 0 (d) x 2 –11x + 60 = 0

Answer: (c) x2 – 3x – 108 = 0

A natural number when increased by 12, equals 160 times its reciprocal. (a) x 2 – 12x + 160 = 0 (b) x 2 – 160x + 12 = 0 (c) 12x 2 – x – 160 = 0 (d) x 2 + 12x – 160 = 0

Answer: (d) x2 + 12x – 160 = 0

2)Nature of Roots A quadratic equation can be defined as an equation of degree 2. This means that the highest exponent of the polynomial in it is 2. The standard form of a quadratic equation is ax 2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. Every quadratic equation has two roots depending on the nature of its discriminant, D = b 2 – 4ac

Which of the following quadratic equation have no real roots? (a) –4x 2 + 7x – 4 = 0 (b) –4x 2 + 7x – 2 = 0 (c) –2x 2 +5x – 2 = 0 (d) 3x 2 + 6x + 2 = 0

Answer: (a) –4×2 + 7x – 4 = 0

Which of the following quadratic equation have rational roots? (a) x 2 + x – 1 = 0 (b) x 2 – 5x + 6 = 0 (c) 4x 2 – 3x – 2 = 0 (d) 6x 2 – x + 11 = 0

Answer: (b) x2 – 5x + 6 = 0

Which of the following quadratic equation have irrational roots? (a) 3x 2 +2x + 2 = 0 (b) 4x 2 – 7x + 3 = 0 (c) 6x 2 – 3x – 5 = 0 (d) 2x 2 +3x – 2 = 0

Answer: (c) 6×2 – 3x – 5 = 0

Which of the following quadratic equations have equal roots? (a) x 2 – 3x + 4 = 0 (b) 2x 2 – 2x + 1 = 0 (c) 5x 2 – 10x + 1 = 0 (d) 9x 2 + 6x + 1 = 0

Answer: (d) 9×2 + 6x + 1 = 0

Which of the following quadratic equations has two distinct real roots? (a) x 2 + 3x + 1 = 0 (b) –x 2 + 3x – 3 = 0 (c) 4x 2 + 8x + 4 = 0 (d) 3x 2 + 6x + 4 = 0

Answer: (a) x2 + 3x + 1 = 0

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Case Study Questions Class 10 Maths Quadratic Equations

Case study questions class 10 maths chapter 4 quadratic equations.

CBSE Class 10 Case Study Questions Maths Quadratic Equations. Term 2 Important Case Study Questions for Class 10 Board Exam Students. Here we have arranged some Important Case Base Questions for students who are searching for Paragraph Based Questions Quadratic Equations.

At Case Study Questions there will given a Paragraph. In where some Important Questions will made on that respective Case Based Study. There will various types of marks will given 1 marks, 2 marks, 3 marks, 4 marks.

CBSE Case Study Questions Class 10 Maths Quadratic Equations

CASE STUDY 1:

Raj and Ajay are very close friends. Both the families decide to go to Ranikhet by their own cars. Raj’s car travels at a speed of x km/h while Ajay’s car travels 5 km/h faster than Raj’s car. Raj took 4 hours more than Ajay to complete the journey of 400 km.

[ CBSE Question Bank ]

4.) How much time took Ajay to travel 400 km?

Answer – d) 16 hour

1.) What will be the distance covered by Ajay’s car in two hours?

a) 2(x +5)km

b) (x – 5)km

c) 2(x + 10)km

d) (2x + 5)km

Answer – a) 2(x +5) km

3.) What is the speed of Raj’s car?

a) 20 km/hour

b) 15 km/hour

c) 25 km/hour

d) 10 km/hour

Answer – a) 20 km/hour

CASE STUDY 2 –

Q.2) Nidhi and Riya are very close friends. Nidhi’s parents have a Maruti Alto. Riya ‘s parents have a Toyota. Both the families decided to go for a picnic to Somnath Temple in Gujarat by their own car. Nidhi’s car travels x km/h, while Riya’s car travels 5km/h more than Nidhi’s car. Nidhi’s car took 4 hours more than Riya’s car in covering 400 km.

[ KVS Raipur 2021 – 22 ]

(i) What will be the distance covered by Riya’s car in two hours? How much time took Riya to travel 400 km?

Answer- 2(x+5)km

(ii) Write the quadratic equation describe the speed of Nidhi’s car. What is the speed of Nidhi’s car?

Answer –  x 2 +5x -500= 0

We hope that above case study questions will help you for your upcoming exams. To see more click below – 

• CBSE Class 10 Maths (standard)
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Study Guides > College Algebra: Co-requisite Course

Quadratic equations, learning objectives.

• Use the zero product principle to solve quadratic equations that can be factored
• Identify solutions to quadratic equations on a graph
• Use the square root property to solve a quadratic equation
• Complete the square to solve a quadratic equation
• Write a quadratic equation in standard form and identify the values of a , b , and c in a standard form quadratic equation.
• Use the Quadratic Formula to find solutions of a quadratic equation, (rational, irrational and complex)

The Zero-Product Property and Quadratic Equations

Solving quadratics with a leading coefficient of 1, reminder: given a quadratic equation with the leading coefficient of 1, factor it..

• Find two numbers whose product equals c and whose sum equals b .
• Use those numbers to write two factors of the form [latex]\left(x+k\right)\text{ or }\left(x-k\right)[/latex], where k is one of the numbers found in step 1. Use the numbers exactly as they are. In other words, if the two numbers are 1 and [latex]-2[/latex], the factors are [latex]\left(x+1\right)\left(x - 2\right)[/latex].
• Solve using the zero-product property by setting each factor equal to zero and solving for the variable.

Answer: Recognizing that the equation represents the difference of squares, we can write the two factors by taking the square root of each term, using a minus sign as the operator in one factor and a plus sign as the operator in the other. Solve using the zero-factor property. [latex]\begin{array}{l}{x}^{2}-9=0\hfill \\ \left(x - 3\right)\left(x+3\right)=0\hfill \\ \hfill \\ \left(x - 3\right)=0\hfill \\ x=3\hfill \\ \hfill \\ \left(x+3\right)=0\hfill \\ x=-3\hfill \end{array}[/latex] The solutions are [latex]x=3[/latex] and [latex]x=-3[/latex].

 Solving Quadratics with a Leading Coefficient of [latex]\ne1[/latex]

• With the quadratic in standard form, [latex]a{x}^{2}+bx+c=0[/latex], multiply [latex]a\cdot c[/latex].
• Find two numbers whose product equals [latex]ac[/latex] and whose sum equals [latex]b[/latex].
• Rewrite the equation replacing the [latex]bx[/latex] term with two terms using the numbers found in step 1 as coefficients of x.
• Factor the first two terms and then factor the last two terms. The expressions in parentheses must be exactly the same to use grouping.
• Factor out the expression in parentheses.
• Set the expressions equal to zero and solve for the variable.

Answer: This equation does not look like a quadratic, as the highest power is 3, not 2. Recall that the first thing we want to do when solving any equation is to factor out the GCF, if one exists. And it does here. We can factor out [latex]-x[/latex] from all of the terms and then proceed with grouping. [latex]\begin{array}{l}-3{x}^{3}-5{x}^{2}-2x=0\hfill \\ -x\left(3{x}^{2}+5x+2\right)=0\hfill \end{array}[/latex] Use grouping on the expression in parentheses. [latex]\begin{array}{l}-x\left(3{x}^{2}+3x+2x+2\right)=\hfill&0\hfill \\ -x\left[3x\left(x+1\right)+2\left(x+1\right)\right]=\hfill&0\hfill \\ -x\left(3x+2\right)\left(x+1\right)=\hfill&0\hfill \end{array}[/latex] Now, we use the zero-product property. Notice that we have three factors. [latex]\begin{array}{l}-x\hfill&=0\hfill \\ x\hfill&=0\hfill \\ 3x+2\hfill&=0\hfill \\ x\hfill&=-\frac{2}{3}\hfill \\ x+1\hfill&=0\hfill \\ x\hfill&=-1\hfill \end{array}[/latex] The solutions are [latex]x=0[/latex], [latex]x=-\frac{2}{3}[/latex], and [latex]x=-1[/latex].

Solve a Quadratic Equation by the Square Root Property

The square root property.

[latex]\begin{array}{l}x^{2}=9\\\,\,\,x=\pm\sqrt{9}\\\,\,\,x=\pm3\end{array}[/latex]

[latex]10x^{2}+5=85[/latex]

[latex]10x^{2}=80[/latex]

[latex] \begin{array}{l}{{x}^{2}}=8\\\,\,\,x=\pm \sqrt{8}\\\,\,\,\,\,\,=\pm \sqrt{(4)(2)}\\\,\,\,\,\,\,=\pm \sqrt{4}\sqrt{2}\\\,\,\,\,\,\,=\pm 2\sqrt{2}\end{array}[/latex]

[latex]\left(x-2\right)^{2}-50=0[/latex]

[latex]\begin{array}{r}\left(x-2\right)^{2}=50\,\,\,\,\,\,\,\,\,\,\\x-2=\pm\sqrt{50}\end{array}[/latex]

[latex] \begin{array}{l}x=2\pm \sqrt{50}\\\,\,\,\,=2\pm \sqrt{(25)(2)}\\\,\,\,\,=2\pm \sqrt{25}\sqrt{2}\\\,\,\,\,=2\pm 5\sqrt{2}\end{array}[/latex]

Solve a Quadratic Equation by Completing the Square

Answer: First notice that the [latex]x^{2}[/latex] term and the constant term are both perfect squares. [latex-display]\begin{array}{l}9x^{2}=\left(3x\right)^{2}\\\,\,\,16=4^{2}\end{array}[/latex-display] Then notice that the middle term (ignoring the sign) is twice the product of the square roots of the other terms. [latex-display]24x=2\left(3x\right)\left(4\right)[/latex-display] A trinomial in the form [latex]r^{2}-2rs+s^{2}[/latex] can be factored as [latex](r–s)^{2}[/latex]. In this case, the middle term is subtracted, so subtract r and s and square it to get [latex](r–s)^{2}[/latex]. [latex-display]\begin{array}{c}\,\,\,r=3x\\s=4\\9x^{2}-24x+16=\left(3x-4\right)^{2}\end{array}[/latex-display]

Steps for Completing The Square

• Given a quadratic equation that cannot be factored, and with [latex]a=1[/latex], first add or subtract the constant term to the right side of the equal sign. [latex]{x}^{2}+4x=-1[/latex]
• Multiply the b term by [latex]\frac{1}{2}[/latex] and square it. [latex]\begin{array}{l}\frac{1}{2}\left(4\right)=2\hfill \\ {2}^{2}=4\hfill \end{array}[/latex]
• Add [latex]{\left(\frac{1}{2}b\right)}^{2}[/latex] to both sides of the equal sign and simplify the right side. We have [latex]\begin{array}{l}{x}^{2}+4x+4=-1+4\hfill \\ {x}^{2}+4x+4=3\hfill \end{array}[/latex]
• The left side of the equation can now be factored as a perfect square. [latex]\begin{array}{l}{x}^{2}+4x+4=3\hfill \\ {\left(x+2\right)}^{2}=3\hfill \end{array}[/latex]
• Use the square root property and solve. [latex]\begin{array}{l}\sqrt{{\left(x+2\right)}^{2}}=\pm \sqrt{3}\hfill \\ x+2=\pm \sqrt{3}\hfill \\ x=-2\pm \sqrt{3}\hfill \end{array}[/latex]
• The solutions are [latex]x=-2+\sqrt{3}[/latex], [latex]x=-2-\sqrt{3}[/latex].

[latex]\begin{array}{r}x^{2}-12x=4\,\,\,\,\,\,\,\,\\b=-12\end{array}[/latex]

[latex]\begin{array}{l}x^{2}-12x+36=4+36\\x^{2}-12x+36=40\end{array}[/latex]

[latex]\left(x-6\right)^{2}=40[/latex]

[latex] x-6=\pm\sqrt{40}[/latex]

[latex] \begin{array}{l}x=6\pm \sqrt{40}\\\,\,\,\,=6\pm \sqrt{4}\sqrt{10}\\\,\,\,\,=6\pm 2\sqrt{10}\end{array}[/latex]

Answer: First, move the constant term to the right side of the equal sign. [latex]{x}^{2}-3x=5[/latex] Identify b .[latex]b=-3[/latex] Then, take [latex]\frac{1}{2}[/latex] of the b term and square it. [latex]\begin{array}{l}\frac{1}{2}\left(-3\right)=-\frac{3}{2}\hfill \\ {\left(-\frac{3}{2}\right)}^{2}=\frac{9}{4}\hfill \end{array}[/latex] Add the result to both sides of the equal sign. [latex]\begin{array}{l}\text{ }{x}^{2}-3x+{\left(-\frac{3}{2}\right)}^{2}=5+{\left(-\frac{3}{2}\right)}^{2}\hfill \\ {x}^{2}-3x+\frac{9}{4}=5+\frac{9}{4}\hfill \end{array}[/latex] Factor the left side as a perfect square and simplify the right side. [latex]{\left(x-\frac{3}{2}\right)}^{2}=\frac{29}{4}[/latex] Use the square root property and solve. [latex]\begin{array}{l}\sqrt{{\left(x-\frac{3}{2}\right)}^{2}}\hfill&=\pm \sqrt{\frac{29}{4}}\hfill \\ \left(x-\frac{3}{2}\right)\hfill&=\pm \frac{\sqrt{29}}{2}\hfill \\ x\hfill&=\frac{3}{2}\pm \frac{\sqrt{29}}{2}\hfill \end{array}[/latex] The solutions are [latex]x=\frac{3}{2}+\frac{\sqrt{29}}{2}[/latex], [latex]x=\frac{3}{2}-\frac{\sqrt{29}}{2}[/latex].

[latex]\begin{array}{c}x^{2}+16x=-64\\b=16\end{array}[/latex]

[latex]\begin{array}{l}x^{2}+16x+64=-64+64\\x^{2}+16x+64=0\end{array}[/latex]

[latex]\left(x+8\right)^{2}=0[/latex]

[latex]x+8=0[/latex]

[latex]x=-8[/latex]

The Quadratic Formula

• First, move the constant term to the right side of the equal sign: [latex]a{x}^{2}+bx=-c[/latex]
• As we want the leading coefficient to equal 1, divide through by a : [latex]{x}^{2}+\frac{b}{a}x=-\frac{c}{a}[/latex]
• Then, find [latex]\frac{1}{2}[/latex] of the middle term, and add [latex]{\left(\frac{1}{2}\frac{b}{a}\right)}^{2}=\frac{{b}^{2}}{4{a}^{2}}[/latex] to both sides of the equal sign: [latex]{x}^{2}+\frac{b}{a}x+\frac{{b}^{2}}{4{a}^{2}}=\frac{{b}^{2}}{4{a}^{2}}-\frac{c}{a}[/latex]
• Next, write the left side as a perfect square. Find the common denominator of the right side and write it as a single fraction: [latex]{\left(x+\frac{b}{2a}\right)}^{2}=\frac{{b}^{2}-4ac}{4{a}^{2}}[/latex]
• Now, use the square root property, which gives [latex]\begin{array}{l}x+\frac{b}{2a}=\pm \sqrt{\frac{{b}^{2}-4ac}{4{a}^{2}}}\hfill \\ x+\frac{b}{2a}=\frac{\pm \sqrt{{b}^{2}-4ac}}{2a}\hfill \end{array}[/latex]
• Finally, add [latex]-\frac{b}{2a}[/latex] to both sides of the equation and combine the terms on the right side. Thus, [latex]x=\frac{-b\pm \sqrt{{b}^{2}-4ac}}{2a}[/latex]

Solving a Quadratic Equation using the Quadratic Formula

• Put the equation in standard form first.
• Identify the coefficients, a , b, and c. Be careful to include negative signs if the bx or c terms are subtracted.
• Carefully substitute the values noted in step 2 into the equation. To avoid needless errors, use parentheses around each number input into the formula.
• Simplify as much as possible.
• Use the [latex]\pm[/latex] in front of the radical to separate the solution into two values: one in which the square root is added, and one in which it is subtracted .
• Simplify both values to get the possible solutions.

[latex]\begin{array}{r}x^{2}+4x=5\,\,\,\\x^{2}+4x-5=0\,\,\,\\\\a=1,b=4,c=-5\end{array}[/latex]

[latex] \begin{array}{r}{{x}^{2}}\,\,\,+\,\,\,4x\,\,\,-\,\,\,5\,\,\,=\,\,\,0\\\downarrow\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\downarrow\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\downarrow\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\a{{x}^{2}}\,\,\,+\,\,\,bx\,\,\,+\,\,\,c\,\,\,=\,\,\,0\end{array}[/latex]

[latex] \begin{array}{l}\\x=\frac{-4\pm \sqrt{{{(4)}^{2}}-4(1)(-5)}}{2(1)}\end{array}[/latex]

[latex]x=\frac{-4\pm\sqrt{16+20}}{2}[/latex]

[latex] x=\frac{-4\pm \sqrt{36}}{2}[/latex]

[latex] x=\frac{-4\pm 6}{2}[/latex]

[latex]\begin{array}{c}x=\frac{-4+6}{2}=\frac{2}{2}=1\\\\\text{or}\\\\x=\frac{-4-6}{2}=\frac{-10}{2}=-5\end{array}[/latex]

[latex]\begin{array}{l}x^{2}-2x=6x-16\\x^{2}-2x-6x+16=0\\x^{2}-8x+16=0\end{array}[/latex]

[latex] \begin{array}{r}{{x}^{2}}\,\,\,-\,\,\,8x\,\,\,+\,\,\,16\,\,\,=\,\,\,0\\\downarrow\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\downarrow\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\downarrow\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\a{{x}^{2}}\,\,\,+\,\,\,bx\,\,\,+\,\,\,\,c\,\,\,\,=\,\,\,0\end{array}[/latex]

[latex]\begin{array}{l}x=\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\\\\x=\frac{-(-8)\pm \sqrt{{{(-8)}^{2}}-4(1)(16)}}{2(1)}\end{array}[/latex]

[latex] x=\frac{8\pm \sqrt{64-64}}{2}[/latex]

[latex] x=\frac{8\pm \sqrt{0}}{2}=\frac{8}{2}=4[/latex]

[latex]\begin{array}{r}x^{2}-2x=6x-16\,\,\,\,\,\\\left(4\right)^{2}-2\left(4\right)=6\left(4\right)-16\\16-8=24-16\,\,\,\,\,\,\\8=8\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\end{array}[/latex]

Quadratic equations with complex solutions

[latex]2x^2+3x+6=0[/latex]

Using the quadratic formula to solve this equation, we first identify a, b, and c.

[latex]a = 2,b = 3,c = 6[/latex]

We can place a, b and c into the quadratic formula and simplify to get the following result:

[latex]x=-\frac{3}{4}+\frac{\sqrt{-39}}{4}, x=-\frac{3}{4}-\frac{\sqrt{-39}}{4}[/latex]

Up to this point, we would have said that [latex]\sqrt{-39}[/latex] is not defined for real numbers and determine that this equation has no solutions.  But, now that we have defined the square root of a negative number, we can also define a solution to this equation as follows.

[latex]x=-\frac{3}{4}+i\frac{\sqrt{39}}{4}, x=-\frac{3}{4}-i\frac{\sqrt{39}}{4}[/latex]

In the following example we will work through the process of solving a quadratic equation with complex solutions. Take note that we be simplifying complex numbers - so if you need a review of how to rewrite the square root of a negative number as an imaginary number, now is a good time.

Use the quadratic formula to solve [latex]{x}^{2}+x+2=0[/latex].

Answer: First, we identify the coefficients: [latex]a=1,b=1[/latex], and [latex]c=2[/latex]. Substitute these values into the quadratic formula. [latex]\begin{array}{l}x\hfill&=\frac{-b\pm \sqrt{{b}^{2}-4ac}}{2a}\hfill \\\hfill&=\frac{-\left(1\right)\pm \sqrt{{\left(1\right)}^{2}-\left(4\right)\cdot \left(1\right)\cdot \left(2\right)}}{2\cdot 1}\hfill \\\hfill&=\frac{-1\pm \sqrt{1 - 8}}{2}\hfill \\ \hfill&=\frac{-1\pm \sqrt{-7}}{2}\hfill \\\hfill&=\frac{-1\pm i\sqrt{7}}{2}\hfill \end{array}[/latex] Now we can separate the expression [latex]\frac{-1\pm i\sqrt{7}}{2}[/latex] into two solutions: [latex-display]-\frac{1}{2}+\frac{ i\sqrt{7}}{2}[/latex-display] [latex-display]-\frac{1}{2}-\frac{ i\sqrt{7}}{2}[/latex-display]   The solutions to the equation are [latex]x=\frac{-1+i\sqrt{7}}{2}[/latex] and [latex]x=\frac{-1-i\sqrt{7}}{2}[/latex] or [latex]x=\frac{-1}{2}+\frac{i\sqrt{7}}{2}[/latex] and [latex]x=\frac{-1}{2}-\frac{i\sqrt{7}}{2}[/latex].

[latex]\begin{array}{l}x^{2}+x=-x-3\\x^{2}+2x+3=0\end{array}[/latex]

[latex]a=1, b=2, c=3[/latex]

Substitute values for a, b, c into the quadratic formula.

[latex]\begin{array}{l}x=\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\\\\x=\frac{-2\pm \sqrt{{{(2)}^{2}}-4(1)(3)}}{2(1)}\end{array}[/latex]

[latex]\displaystyle x=\frac{-2\pm \sqrt{-8}}{2}[/latex]

Rewrite the radical of a negative number in terms of the imaginary unit [latex]i[/latex]

[latex]\displaystyle x=\frac{-2\pm i\sqrt{8}}{2}[/latex]

[latex]\displaystyle x=\frac{-2\pm 2i\sqrt{2}}{2}[/latex]

[latex]x=-1 \pm i\sqrt{2}[/latex]

Licenses & Attributions

Cc licensed content, original.

• Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution .
• Quadratic Formula Application - Time for an Object to Hit the Ground. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution .
• Quadratic Formula Application - Determine the Width of a Border. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution .

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• College Algebra. Provided by: OpenStax Authored by: Abramson, Jay, et al.. Located at: https://cnx.org/contents/ [email protected] :1/Preface. License: CC BY: Attribution . License terms: Download for free at : http://cnx.org/contents/ [email protected] :1/Preface.
• Ex: Solve a Quadratic Equation Using Factor By Grouping. Authored by: James Sousa (Mathispower4u.com) . License: CC BY: Attribution .
• Ex: Factor and Solve Quadratic Equation - Trinomial a = -1. Authored by: James Sousa (Mathispower4u.com) . License: CC BY: Attribution .
• Unit 12: Factoring, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology and Education Located at: https://www.nroc.org/. License: CC BY: Attribution .
• Ex 1: Solving Quadratic Equations Using Square Roots. Authored by: James Sousa (Mathispower4u.com) . License: CC BY: Attribution .
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• Ex 2: Completing the Square - Real Irrational Solutions. Authored by: James Sousa (Mathispower4u.com) . License: CC BY: Attribution .
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Case Study on Quadratic Equations Class 10 Maths PDF

The passage-based questions are commonly known as case study questions. Students looking for Case Study on Quadratic Equations Class 10 Maths can use this page to download the PDF file. 

The case study questions on Quadratic Equations are based on the CBSE Class 10 Maths Syllabus, and therefore, referring to the Quadratic Equations case study questions enable students to gain the appropriate knowledge and prepare better for the Class 10 Maths board examination. Continue reading to know how should students answer it and why it is essential to solve it, etc.

Case Study on Quadratic Equations Class 10 Maths with Solutions in PDF

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Why Solve Quadratic Equations Case Study Questions on Class 10 Maths?

There are three major reasons why one should solve Quadratic Equations case study questions on Class 10 Maths - all those major reasons are discussed below:

• To Prepare for the Board Examination: For many years CBSE board is asking case-based questions to the Class 10 Maths students, therefore, it is important to solve Quadratic Equations Case study questions as it will help better prepare for the Class 10 board exam preparation.
• Develop Problem-Solving Skills: Class 10 Maths Quadratic Equations case study questions require students to analyze a given situation, identify the key issues, and apply relevant concepts to find out a solution. This can help CBSE Class 10 students develop their problem-solving skills, which are essential for success in any profession rather than Class 10 board exam preparation.
• Understand Real-Life Applications: Several Quadratic Equations Class 10 Maths Case Study questions are linked with real-life applications, therefore, solving them enables students to gain the theoretical knowledge of Quadratic Equations as well as real-life implications of those learnings too.

How to Answer Case Study Questions on Quadratic Equations?

Students can choose their own way to answer Case Study on Quadratic Equations Class 10 Maths, however, we believe following these three steps would help a lot in answering Class 10 Maths Quadratic Equations Case Study questions.

• Read Question Properly: Many make mistakes in the first step which is not reading the questions properly, therefore, it is important to read the question properly and answer questions accordingly.
• Highlight Important Points Discussed in the Clause: While reading the paragraph, highlight the important points discussed as it will help you save your time and answer Quadratic Equations questions quickly.
• Go Through Each Question One-By-One: Ideally, going through each question gradually is advised so, that a sync between each question and the answer can be maintained. When you are solving Quadratic Equations Class 10 Maths case study questions make sure you are approaching each question in a step-wise manner.

What to Know to Solve Case Study Questions on Class 10 Quadratic Equations?

 A few essential things to know to solve Case Study Questions on Class 10 Quadratic Equations are -

• Basic Formulas of Quadratic Equations: One of the most important things to know to solve Case Study Questions on Class 10 Quadratic Equations is to learn about the basic formulas or revise them before solving the case-based questions on Quadratic Equations.
• To Think Analytically: Analytical thinkers have the ability to detect patterns and that is why it is an essential skill to learn to solve the CBSE Class 10 Maths Quadratic Equations case study questions.
• Strong Command of Calculations: Another important thing to do is to build a strong command of calculations especially, mental Maths calculations.

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Quadratic Equations

An example of a Quadratic Equation :

The function can make nice curves like this one:

The name Quadratic comes from "quad" meaning square, because the variable gets squared (like x 2 ).

It is also called an "Equation of Degree 2" (because of the "2" on the x )

Standard Form

The Standard Form of a Quadratic Equation looks like this:

• a , b and c are known values. a can't be 0
• x is the variable or unknown (we don't know it yet)

Here are some examples:

Have a Play With It

Play with the Quadratic Equation Explorer so you can see:

• the function's graph, and
• the solutions (called "roots").

Hidden Quadratic Equations!

As we saw before, the Standard Form of a Quadratic Equation is

But sometimes a quadratic equation does not look like that!

For example:

How To Solve Them?

The " solutions " to the Quadratic Equation are where it is equal to zero .

They are also called " roots ", or sometimes " zeros "

There are usually 2 solutions (as shown in this graph).

And there are a few different ways to find the solutions:

Just plug in the values of a, b and c, and do the calculations.

We will look at this method in more detail now.

About the Quadratic Formula

First of all what is that plus/minus thing that looks like ± ?

The ± means there are TWO answers:

x = −b + √(b 2 − 4ac) 2a

x = −b − √(b 2 − 4ac) 2a

Here is an example with two answers:

But it does not always work out like that!

• Imagine if the curve "just touches" the x-axis.
• Or imagine the curve is so high it doesn't even cross the x-axis!

This is where the "Discriminant" helps us ...

Discriminant

Do you see b 2 − 4ac in the formula above? It is called the Discriminant , because it can "discriminate" between the possible types of answer:

• when b 2 − 4ac is positive, we get two Real solutions
• when it is zero we get just ONE real solution (both answers are the same)
• when it is negative we get a pair of Complex solutions

Complex solutions? Let's talk about them after we see how to use the formula.

Using the Quadratic Formula

Just put the values of a, b and c into the Quadratic Formula, and do the calculations.

Example: Solve 5x 2 + 6x + 1 = 0

Answer: x = −0.2 or x = −1

Let's check the answers:

Remembering The Formula

A kind reader suggested singing it to "Pop Goes the Weasel":

Try singing it a few times and it will get stuck in your head!

Or you can remember this story:

x = −b ± √(b 2 − 4ac) 2a

"A negative boy was thinking yes or no about going to a party, at the party he talked to a square boy but not to the 4 awesome chicks. It was all over at 2 am. "

Complex Solutions?

When the Discriminant (the value b 2 − 4ac ) is negative we get a pair of Complex solutions ... what does that mean?

It means our answer will include Imaginary Numbers . Wow!

Example: Solve 5x 2 + 2x + 1 = 0

Answer: x = −0.2 ± 0.4 i

The graph does not cross the x-axis. That is why we ended up with complex numbers.

In a way it is easier: we don't need more calculation, we leave it as −0.2 ± 0.4 i .

Example: Solve x 2 − 4x + 6.25 = 0

Answer: x = 2 ± 1.5 i

BUT an upside-down mirror image of our equation does cross the x-axis at 2 ± 1.5 (note: missing the i ).

Just an interesting fact for you!

• Quadratic Equation in Standard Form: ax 2 + bx + c = 0
• Quadratic Equations can be factored
• Quadratic Formula: x = −b ± √(b 2 − 4ac) 2a
• positive, there are 2 real solutions
• zero, there is one real solution
• negative, there are 2 complex solutions

CBSE Case Study Questions for Class 10 Maths Quadratic Equation Free PDF

Mere Bacchon, you must practice the CBSE Case Study Questions Class 10 Maths Quadratic Equation  in order to fully complete your preparation . They are very very important from exam point of view. These tricky Case Study Based Questions can act as a villain in your heroic exams!

I have made sure the questions (along with the solutions) prepare you fully for the upcoming exams. To download the latest CBSE Case Study Questions , just click ‘ Download PDF ’.

CBSE Case Study Questions for Class 10 Maths Quadratic Equation PDF

Checkout our case study questions for other chapters.

• Chapter 2: Polynomials Case Study Questions
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Case Study Questions for Class 10 Maths Chapter 4 Quadratic Equations

• Last modified on: 10 months ago
• Reading Time: 4 Minutes

Question 1:

Raj and Ajay are very close friends. Both the families decide to go to Ranikhet by their own cars. Raj’s car travels at a speed of x km/h while Ajay’s car travels 5 km/h faster than Raj’s car. Raj took 4 h more than Ajay to complete the journey of 400 km.

(i) What will be the distance covered by Ajay’s car in two hours? (a) 2 (x + 5) km (b) (x – 5) km (c) 2 (x + 10) km (d) (2x + 5) km

(ii) Which of the following quadratic equation describe the speed of Raj’s car? (a) x 2 − 5x − 500 = 0 (b) x 2 + 4x − 400 = 0 (c) x 2 + 5x − 500 = 0 (d) x 2 − 4x + 400 = 0

(iii) What is the speed of Raj’s car? (a) 20 km/h (b) 15 km/h (c) 25 km/h (d) 10 km/h

(iv) How much time took Ajay to travel 400 km? (a) 20 h (b) 40 h (c) 25 h (d) 16 h

(v) How much time took Raj to travel 400 km? (a) 15 h (b) 20 h (c) 18 h (d) 22 h

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PUMPA - SMART LEARNING

எங்கள் ஆசிரியர்களுடன் 1-ஆன்-1 ஆலோசனை நேரத்தைப் பெறுங்கள். டாப்பர் ஆவதற்கு நாங்கள் பயிற்சி அளிப்போம்

• Mathematics CBSE
• Quadratic Equations
• Nature of Roots

12. Case study: Nature of roots

Exercise condition:.

• 2 x 2 − 4 x + 390 = 0
• 2 x 2 − 4 x − 390 = 0
• 2 x 2 + 4 x − 390 = 0
• 2 x 2 + 4 x + 390 = 0
• \(ax^2 - bx + c = 0\)
• \(ax^2 + bx + c = 0\)
• \(ax^2 + bx = c\)
• \(ax^2 + ax + c = 0\)
• \(b^2 - 4ac = 0\)
• \(b^2 - 4ac \le 0\)
• \(b^2 - 4ac < 0\)
• \(b^2 - 4ac > 0\)

Plus.Maths.org

101 uses of a quadratic equation

It isn't often that a mathematical equation makes the national press, far less popular radio, or most astonishingly of all, is the subject of a debate in the UK parliament. However, in 2003 the good old quadratic equation, which we all learned about in school, was all of those things.

Where we begin

Where would it all end? Was the quadratic equation really dead? Did anyone care? Are mathematicians really evil monsters who only want to inflict quadratic equations on a younger generation as a means of corrupting their immortal souls?

Maybe so, but it's not really the quadratic equation's fault. In fact, the quadratic equation has played a pivotal part in not only the whole of human civilisation as we know it, but in the possible detection of other alien civilisations and even such vital modern activities as watching satellite television. What else, apart from the nature of divine revelation, could be considered to have had such an impact on life as we know it? Indeed, in a very real sense, quadratic equations can save your life.

The Babylonians

Babylonian cuneiform tablets recording the 9 times tables

Finding square roots by using a calculator is easy for us, but was more of a problem for the Babylonians. In fact they developed a method of successive approximation to the answer which is identical to the algorithm (called the Newton-Raphson method ) used by modern computers to solve much harder problems than quadratic equations.

Now we complete the square by using the fact that

Combining this with the original equation we have

which can be rewritten as

The fact that taking a square root can give a positive or a negative answer leads to the remarkable result that a quadratic equation has two solutions. So much for mathematical puzzles only having one solution!

A surprise for the Greeks, a bit of mathematical origami and a sense of proportion

They knew that this equation had a solution. In fact it is the length of the hypotenuse of a right angled triangle which had sides of length one.

Paper used in the United States, called foolscap , has a different proportion. To see why, we return to the Greeks and another quadratic equation. Having caused such grief, the quadratic equation redeems itself in the search for the perfect proportions: a search that continues today in the design of film sets, and can be seen in many aspects of nature.

This is yet another quadratic equation : a very important one that comes up in all sort of applications. It has the (positive) solution

In this sequence each term is the sum of the previous two terms. Fibonacci discovered it in the 15th century in an attempt to predict the future population of rabbits. If you take the ratio of each term to the one after it, you get the sequence of numbers

Conics link quadratic equations to the stars

Half of the cone can be visualised as the spread of light coming from a torch. Now, if you shine a torch onto a flat surface such as a wall then you will see various shapes as you move the torch around. These shapes are called conic sections and are the curves that you obtain if you take a slice through a cone at various different angles. Precisely these curves were studied by the Greeks, and they recognised that there were basically four types of conic section. If you take a horizontal section through the cone then you get a circle . A section at a small angle to the horizontal gives you an ellipse . If you take a vertical section then you get a hyperbola and if you take a section parallel to one side of the cone then you get a parabola . These curves are illustrated below.

These curves were known and studied since the Greeks, but apart from the circle they did not seem to have any practical application. However, as we shall see in the next issue of Plus , a link between quadratic equations and conics, coupled with a mighty lucky fluke, led to an understanding of the way that the universe worked, and in the 16th century the time came for conics to change the world.

About the authors

Chris Budd is Professor of Applied Mathematics in the Department of Mathematical Sciences at the University of Bath, and the Chair of Mathematics at the Royal Institution in London.

Chris Sangwin is a member of staff in the School of Mathematics and Statistics at the University of Birmingham. He is a Research Fellow in the Learning and Teaching Support Network centre for Mathematics, Statistics, and Operational Research.

They have recently written the popular mathematics book Mathematics Galore! , published by Oxford University Press.

This article was inspired in part by a remarkable debate in the British House of Commons on the subject of quadratic equations. The record of this debate can be found in Hansard, United Kingdom House of Commons, 26 June 2003, Columns 1259-1269, 2003, which is available online at the House of Commons Hansard Debate website.

• Add new comment

Its all in the context

What a great shame such concepts are seldom if ever taught in schools framed in historical and application context as they have been in this article.

And also the description

I agree with you. Children also need to know that a quadratic equation is a way to describe a physical thing, such as the arch of a bridge. I never understood that and no one ever told me that. I figured it out on my own when, as an adult, I restudied math. If they could see that math is a descriptive language that we all make use of, whether we fully realize it or not, they would then A) more readily understand math, and B) see its worth rather than question its worth.

Babylonians?

Just nitpicking here, but wouldn't 3000 B.C. be the Sumerians, not the Babylonians? It's been a while since I took Near Eastern studies, but I'm pretty sure the Babylonians didn't come around until mid third millenium B.C.

Yes you are right, they weren

Yes you are right, they weren't around yet.

thank you!!!

The information regarding quadratic formulas has helped me visualize and clearly grasp the concept for potential applications. Amazing!! I am one who has struggled in the subject yet determined to understand it.

Very Cool Story!

Hey, I really like your website. For the first time ever, I see the importance of the quadratic formula. However, I need some help with a step, using the triangular field to develop c = ax^2 + bx. Why is the height 2x for the two right triangles. It's probably easy, but it's a step that is mysterious to me and my long work day, tired brain.

Triangular field area

Great article, wonderful introduction to quadratic equations.

I had exactly the same difficulty - and after over 50 years my school maths is really, really rusty.

There must be a good reason as to why they chose those particular ways of denoting base and height of the triangle (rather than just b and h), and I assume it is correct. Still do not really understand that bit.

However if you use those expressions in the area formula for a triangle (area = half base time height) then it does come out.

Also stumped

I could follow through up to the triangular field example. I suspect that there is some factoring and cancelling going on.

Nevertheless, I found this article useful in explaining WHY we have quadratic equations, and knowing why we have them helps me understand why/where/how we might apply them in real life situations. This was never taught to me at school. Instead, we just had to recognise when an equation was quadratic, learn the formula, then apply it.

Re: Very Cool Story! Thanks for your rectifying.

Yes, the formula that follows now makes sense. The height of the perpendicular now makes sense where it is "2x", and not "2x/m".

Appreciation

Wonderful article!!!!!!!!!

101 things Quadratic

I enjoyed this article very much. The combination of history, that I love, with algebra, that I struggle with, made my understanding of the concepts of functions easier. I especially appreciated seeing the conics sections and the application of each into graph equations. You all gave me an "aaha" moment. Keep up the good work. Debra

that's so first grade

I learned to factor quadratics in first grade. They're tame. I learned how functions can model "anything" when i watched the standard deviants algebra videos.

More generalized polynomials can be a pain to factor, though.

Well, okay..

Well, you must have been the smartest first grader in the world, I can't even figure this out in the 8th grade.

It must have been a great effort to bring out these significance of quadratic equations. Immensely impressed. Appreciate the good job..

the research done on quadratic equation is awesome i always found it boooooring but this what is written is simply great. the Babylonians and the Greeks are awesome i really loved reading it and hence forth in 11std i would surely do it thoroughly.thank you very much authors

INTERESTING

I always said that had no meaning at all, and why learn it if I won't ever use it again. This article has completely shut me up. I enjoyed every bit of this arcticle, very interesting introduction on Quadratic Equations. The information providded on quadratics had seriously helped me understand it a lot more. Its amazing how they use physical things such as the bridge and the arch to solve the dimensions.

United States needs to change mathematics instruction from K-12

This helped me understand the relevance of the quadratic equation. Incorporating the history of mathematics demonstrates how mathematics helps people become more efficient to finding solutions to world problems. Many students shut down mentally and emotionally when it comes to mathematics. in the United States. I'm trying to find ways to help change the way we instruct in the United States.

Bad teachers

I don't think that I ever had a math instructor that actually knew the subject until I reached college. It was a true joy to ask questions and get real answers. The US is crippled in math and science because k-12 education has become a union racket to employ the otherwise useless. The best way to change the way we instruct is to abolish all state funded public schools, disband public unions that kick back campaign money to the supposed representatives and let the parents and local school boards freely fire the worthless drones.

Math Teachers and Low pay

Actually, the reason why we can't get good math teachers is becuase the industry hires them at a much higher rate of pay then what the schools can pay. We get the "left overs" to choose from. I lucked out, and happened to get 3 very good math teachers. But I was the exception, and clearly not the rule.

Mirtha Abreu - Use of Quadratic Equation

This has definitely helped me understand quadratic equations. This is a subject that I have previously struggles with an after reading this article, I can understand it much better. I enjoyed learning about the history of quadratic equations and reading the explanations. Great article and very well put out!

Word Confusion

Part of the Quadratic Equation Article states:

"which is in turn proportional to the square of the length of the side. In mathematical terms, if (x) is the length of the side of the field, (m) is the amount of crop you can grow on a square field of side length 1, and (c) is the amount of crop that you can grow, then"

"m" and "c" sound like the same thing? Is this a typo?

The two are different: m is

The two are different: m is the amount you can grow on a field of unit side length and c the amount you can grow on the field under consideration (side length x).

Too really help...expand more on the triangle

Please expand on how you derived the labels on the Triangle and how then they fulfill the equation c = ax^2 + bx.

I still don't like the Irrational ones though...

"At this point the Greeks gave up algebra and turned to geometry."

Honestly? So did I! I am an artist, I think graphically. Geometry, Geography, Cartography, Orthography, etc. have always come to me easily. Irrational Quadratic Equations (IQE), as taught in most public schools in the United States of America, make absolutely no sense, and serve no discernible purpose in the real world.

My own instructors dedicated 50% or more of their courses to IQE, frustrating me to no end, because they wouldn't move on to anything else once they reached them. They constantly asked on written assignments to merely, "Solve.", equations. Then they always complained about the result I wrote, even when it was correct, because they wanted me to, "Show my work."

The process of going through the formula was more important to them than the result. None of them understood that I used a different means to get to the result, that was faster, and just as accurate. I didn't understand why they insisted upon writing mathematical expressions that were needlessly complex to denote an equation that was effectively upside down, backwards, and turned inside out. For them, algebraic notation was a mathematical puzzle to be taken apart and put back together, providing 'proof' that the expression was true at all points in the progression.

I skipped the algebraic notation and went directly to the result. I didn't need 'proof', I just wanted to get the work done. I knew in my heart that no one would actually write equations of the sort they expressed when attempting to solve real world issues in an expedited manner.

This article is very well written. I wish I had come across something of this sort thirty years ago, when it could have done me some good. Instead, it wasn't until I took classes in Trigonometry that it all fell into place. Trigonometry did for me, as an artist, what Algebra did for my high school instructors. Trigonometry acted as a mathematical bridge between Arithmetic, Geometry and Algebra, that I could traverse at will.

Unta Glebin Gloutin Globin

Red Ronin, The Cybernetic Samurai

360 degrees, 365.25 days

I think it is nearly impossible that the Babylonians thought there were 360 days in a year. I think you are implying that the number of degrees in a circle were chosen because the earth moves through almost one degree of its orbit each day. It's more likely that they chose 360 degrees as an outgrowth of their love for the number 60 - because it has so many factors. If you choose 60 for the internal angle of an equilateral triangle you get 360 degrees in a circle.

The radius of a circle will fit inside the circle six times exactly to form a hexagon; the corners of the hexagon each touch the circumference of the circle. Babylonians did indeed have a love for the number 60 and if each of the sides of the hexagon are divided into 60 and a line drawn from each 60th to the centre of the circle then there are 360 divisions in the circle.

Much appreciated

Thanks for going to the trouble of explaining the history and applications of quadratic equations. The point of it all was never explained to me when I was thrown into the deep end with them, age 10. Now that I've been asked to explain them to a friend's son, your material is helping to demystify things. Matt, North Wales, UK

Don't understand how you get to c = a x^2 + b x

I can't get beyond c = a x^2 + b x. How is this equation derived from the figure given? There's no explanation as to what "a" and "b" actually represent?

re: Don't understand how you get to c = a x^2 + b x

I was wondering the same thing. In the diagram I take ax to be the base of the smaller triangle but then where is x in the equation coming from? Are a and x equal?

Also don't understand how that 1st 2 triangles field eqn works

I'm also stuck on that 1st example of the field comprising 2 triangles and how we get to the quadratic equation from that. I would love to go through the rest of this article but don't want to until I've overcome the hurdle of understanding this. Please, someone?

The area of the smaller of

The area of the smaller of the two triangles is ax^2/m and the area of the larger one is bx/m (from the standard formula for the area of triangles). This means that the area of the whole field is ax^2/m+bx/m. Since the amount of crop that can be grown on a field of unit area is m, the amount of crop that can be grown on a field of area ax^2/m+bx/m is m(ax^2/m+bx/m) = ax^2+bx.

Area of smaller triangle

But why is the base of one triangle ax and of the other simply b. Where does that ax value come from?

can someone please explain terrys question-why is one base ax and the other one simply b. also why is the height 2x/m. where does the m come from.

Explanations

I can understand Anon's frustration back in Jan '16. So often in mathematical explanations I've read I find myself tripping over a missing step. Like a mathematical pothole. It's usually something so obvious to the mathematician who wrote it that it didn't seem to need mentioning. ( Like where that little square came from- though I did eventually work that one out). The problem is that if you are trying to follow a set of mathematical steps even if you solve the missing one (as with me and the small square) you have been diverted away from the main problem and lost the thread: And then probably give up and go off and do something else instead.

It's just a quadratic equation in standard form tweaked ie ax^2 + bx + c vs c = ax^2 + bx and if a is anything other than 1, then you remove it, etc~

Will try to help you clarify

I'm pretty sure when they sought out ways to derive a quadratic equation to help them reason triangular regions they had to think frontwards and backwards. First, I believe you need to understand how the height 2x/m came into play (why it was used). First, keep in mind that "m" represents a basic unit of 1. That would mean that 2x/m (the height for BOTH triangles would appear to be 2x. But are their heights 2x? Let's think about it, when finding the area of a right triangle we eventually divide the area by 2 after multiplying the b x the h. Knowing this, it is mathematically reasonable why the coefficient of "2" was put in front of x-it would get divided back out and preserve what they really wanted for the height of the triangle/ length of one side of the land "x". This mean that the small triangular area would be ax times x or (ax^2). The larger triangular area would be b times x or bx for its area. You asked though "what is "a" and "b"? look at length of base "a", compared to the triangles height that we previously deduced to really be "x". "a" represents a coefficient thats taking a fraction of base length "x" for the small base is being represented in terms of the height of the triangle or length of the land. Base b ls obviously the second width of the scalene triangle or width of land that IF represented with bx instead of b (like it is) would have created a bx^2 term instead of the bx we need to figure out the area the land in addition to other things. This is my perception after being confused there for a minute too. I hope this helped you or someone just a little although it's years later- just discovered this awesome forum:).

Super Confused

I really can't follow what you're saying. I just want to know where that expression for the height comes from. So I called it h to get the total area of the triangle as h(ax+b)/2. Total total yield of this area will be hm(ax + b)/2. So I can appreciate that ax must be something relating that smaller triangle to the height, and if I set h = x, I get m(ax^2 + bx)/2 for the total yield. Substituting h = 2x/m gives ax^2 + bx which is the area of two quadrilaterals with the same height of x and 2 sides of ax and b. So the yield, which should be a product of area and the coefficient m is now rendered as the areas of two squares without having anything to do with that coefficient anymore. Taking the height to be x again and the bases as they are, the total yield for the aggregate quadrilateral is m(ax^2 + bx), and for the triangles would be that over 2. Rearranging gives ax^2 + bx = 2yield /m. So if the yield of the quadrilateral (divided by m) of height x is ax^2 + bx, then the height at which the yeild of the triangles is equal to that is 2x/m. I can see all that but I just can't grasp what on earth is going on and its doing my head in

Sir, the Babylonians were not

Sir, the Babylonians were not in power in 3000 BCE, it was the Sumerians. Babylonians took over Mesopotamia at around 1900 BCE.

Please provide a link to the next issue continue this topics on conics and quandraic.

Great!!!! what an explanation

Great!!!! what an explanation.. really we need a teacher who teaches basics like this in the beginning to all children

Thanks so much I kept getting my anwsers wrong because I didn't realize you had to divide both parts by the denominato

“The reason that we teach

“The reason that we teach symbolic algebraic manipulation has everything to do with its efficiency and nothing to do with the historical development of the problems being solved.” -- Patricia R. Allaire and Robert E. Bradley (in their article ‘Geometric Approaches to Quadratic Equations from Other Times and Places’, available online at: https://www.maa.org/sites/default/files/images/upload_library/46/NCTM/G… )

Why the height is 2x/m

I noticed a few people were confused about the choice of height for the triangle, so here is my explanation :) m is the amount of crops that you can grow in 1 square unit of area. c = mx*x is the amount of crops you can grow in a square field of length x. So, c = m*Area of Square For triangle, c = m* Area of Triangle = m (base*height)/2 Base = ax*x + b*x Choose height = 2x/m so that: c = m (ax*x + b)*(2x/m)/2 the m on the numerator is divided by m on the denominator to give 1, and 2/2 also gives 1 So, height = 2x/m means that the m "cancels out" and so does the 2 to give the desired quadratic formula: c = ax*x + b*x

Hope this helps.

Where did "m" comes from

Thanks for your comment, it has been helpful.

But I would like to know why did they use (2x/m) as its height? And I would like to break that question into two

1) having ax on the small triangle's base implies that there's a relationship with the base and it's height. What is the relationship? Isn't it two separate events?

2) Similar question to "m" how does the "m" in the height have any relation with the "m" that's being used for amount of crops that can grow in a sqft.

Because if the height was replaced with a "y", the whole formula wouldn't exist eventho it may or may not equals to 2b/m.

Oooops. Typo. My previous explanation should have "ax" not "ax*x" :)

Oh dear. The "b*x" should be just a "b". Way past my bedtime :)

Quadratic Function and Torricelli's fountain

Esteemed Budd and Sungwin, Your collection of examples of the use of the quadratic equation is Excellent. My congratulations to you for this collection. We found it while writing our article “Visualizing Properties of a Quadratic Function Using Torricelli’s Fountain” which was published in The Physics Teacher. We cited your collection in our article. Mirjana Božić

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• Quadratic Equation For Class 10

Quadratic Equation Class 10 Notes Chapter 4

Cbse class 10 maths quadratic equation notes:- download pdf here.

Get the complete concepts covered in quadratic equations for Class 10 Maths here. These quadratic equations notes help the students to recall the important definitions, formulas and tricks to solve the problems in the CBSE Board Exams 2023-24. In this article, you will learn the concept of quadratic equations, standard form, nature of roots, and methods for finding the solution for the given quadratic equations with more examples.

Introduction to Quadratic Equations

Quadratic polynomial.

A polynomial of the form a x 2 + b x + c , where a, b and c are real numbers and a ≠ 0 is called a quadratic polynomial.

For more information on Quadratic Polynomials, watch the below video

Quadratic Equation

When we equate a quadratic polynomial to a constant, we get a quadratic equation.

Any equation of the form p(x) = k, where p(x) is a polynomial of degree 2, and c is a constant, is a quadratic equation.

The Standard Form of a Quadratic Equation

The standard form of a quadratic equation is ax 2 +bx+c=0, where a,b and c are real numbers and a≠0. ‘a’ is the coefficient of x 2 . It is called the quadratic coefficient. ‘b’ is the coefficient of x. It is called the linear coefficient. ‘c’ is the constant term.

To know more about Quadratic Equations, visit here .

Solving Quadratic Equations by Factorisation

Roots of a quadratic equation.

The values of x for which a quadratic equation is satisfied are called the roots of the quadratic equation.

If α is a root of the quadratic equation ax 2 +bx+c=0, then aα 2 +bα+c=0.

A quadratic equation can have two distinct real roots, two equal roots or real roots may not exist.

Graphically, the roots of a quadratic equation are the points where the graph of the quadratic polynomial cuts the x-axis.

Consider the graph of a quadratic equation x 2 −4=0

In the above figure, -2 and 2 are the roots of the quadratic equation x 2 −4=0 Note:

• If the graph of the quadratic polynomial cuts the x-axis at two distinct points, then it has real and distinct roots.
• If the graph of the quadratic polynomial touches the x-axis, then it has real and equal roots.
• If the graph of the quadratic polynomial does not cut or touch the x-axis then it does not have any real roots.

Solving a Quadratic Equation by Factorization Method

Consider a quadratic equation 2x 2 −5x+3=0

⇒2x 2 −2x−3x+3=0

This step is splitting the middle term.

We split the middle term by finding two numbers (-2 and -3) such that their sum is equal to the coefficient of x and their product is equal to the product of the coefficient of x 2 , and the constant.

(-2) + (-3) = (-5)

And (-2) × (-3) = 6

2x 2 −2x−3x+3=0

2x(x−1)−3(x−1)=0

(x−1)(2x−3)=0

In this step, we have expressed the quadratic polynomial as a product of its factors.

Thus, x = 1 and x =3/2 are the roots of the given quadratic equation.

This method of solving a quadratic equation is called the factorisation method.

For more information on Solving a Quadratic Equation by Factorization Method, watch the below video

To know more about Solving Quadratic Equation by Factorisation, visit here .

Solving a Quadratic Equation by Completion of Squares Method

In the method of completing the squares, the quadratic equation is expressed in the form (x±k) 2 =p 2 .

Consider the quadratic equation 2x 2 −8x=10 (i) Express the quadratic equation in standard form. 2x 2 −8x−10=0

(ii) Divide the equation by the coefficient of x 2 to make the coefficient of x 2 equal to 1. x 2 −4x−5=0

(iii) Add the square of half of the coefficient of x to both sides of the equation to get an expression of the form x 2 ±2kx+k 2 . (x 2 −4x+4)−5=0+4

(iv) Isolate the above expression, (x±k) 2 on the LHS to obtain an equation of the form (x±k) 2 =p 2 (x−2) 2 =9

(v) Take the positive and negative square roots. x−2=±3

x=−1 or x=5

To know more about Solving Quadratic Equations by Completing the Square, visit here .

Solving Quadratic Equation Using Quadratic Formula

Quadratic formula.

Quadratic Formula is used to directly obtain the roots of a quadratic equation from the standard form of the equation.

For the quadratic equation ax 2 +bx+c=0,

x= [-b± √(b 2 -4ac)]/2a

By substituting the values of a,b and c, we can directly get the roots of the equation.

Example: If x 2 – 5x + 6 = 0 is the quadratic equation, find the roots.

Solution: Given, x 2 – 5x + 6 = 0 is the quadratic equation.

On comparing with the standard quadratic equation, we have;

ax 2 + bx + c = 0

a = 1, b = -5 and c = 6

b 2 – 4ac = (-5) 2 – 4 × 1 × 6 = 25 – 24 = 1 > 0 Hence, the roots are real. Using quadratic formula,

x = [-b ± √(b 2 – 4ac)]/ 2a

= [-(-5) ± √1]/ 2(1)

= [5 ± 1]/ 2

i.e. x = (5 + 1)/2 and x = (5 – 1)/2

x = 6/2, x = 4/2

Therefore, the roots of the quadratic equation are 3 and 2.

To know more about Quadratic Formula, visit here .

Discriminant

For a quadratic equation of the form ax 2 +bx+c=0, the expression b 2 −4ac is called the discriminant (denoted by D) of the quadratic equation.

The discriminant determines the nature of the roots of the quadratic equation based on the coefficients of the quadratic equation.

For more information on Discriminant, watch the below video

To know more about Discriminant Formula, visit here .

Nature of Roots

Based on the value of the discriminant, D=b 2 −4ac, the roots of a quadratic equation, ax 2 + bx + c = 0, can be of three types.

Case 1: If D>0 , the equation has two distinct real roots .

Case 2: If D=0 , the equation has two equal real roots .

Case 3: If D<0 , the equation has no real roots .

Solving using Quadratic Formula when D>0

Solve 2x 2 −7x+3=0 using the quadratic formula.

(i) Identify the coefficients of the quadratic equation. a = 2,b = -7,c = 3

(ii) Calculate the discriminant, b 2 −4ac

D=(−7) 2 −4×2×3= 49-24 = 25

D> 0, therefore, the roots are distinct.

(iii) Substitute the coefficients in the quadratic formula to find the roots

x= [-(-7)± √((-7) 2 -4(2)(3))]/2(2)

x=3 and x= 1/2 are the roots.

Solving Quadratic Equation when D=0

Let us take an example of quadratic equation 3x 2 – 2x + 1/3 = 0.

Here, a = 3, b = -2 and c = 1/3

Determinant, D = b 2 – 4ac = (-2) 2 – 4 (3)(1/3) = 4 – 4 = 0

Thus, the given equation has equal roots.

Hence the roots are -b/2a and -b/2a, i.e., 1/3 and 1/3.

Solving Quadratic Equation when D < 0

Suppose the quadratic equation is 4x 2 + 3x + 5 = 0

Comparing with the standard form of quadratic equation, ax 2 + bx + c = 0,

a = 4, b = 3, c = 5

By the formula of determinant, we know;

Determinant (D) = b 2 – 4ac

= (3) 2 – 4(4)(5)

= -71 2 – 4ac)]/ 2a

= [-3 ± √(-71)]/ 2(4)

= [-3 ± √(i 2 71)]/ 8

= (-3 ± i√71)/8

Thus, the non-real roots of the equation are x = (-3 + i√71)/8 and x (-3 – i√71)/8.

For more information on Nature Of Roots, watch the below videos

To know more about the Nature of Roots, visit here .

Graphical Representation of a Quadratic Equation

The graph of a quadratic polynomial is a parabola. The roots of a quadratic equation are the points where the parabola cuts the x-axis i.e. the points where the value of the quadratic polynomial is zero.

Now, the graph of x 2 +5x+6=0 is:

In the above figure, -2 and -3 are the roots of the quadratic equation x 2 +5x+6=0.

For a quadratic polynomial ax 2 +bx+c,

If a>0, the parabola opens upwards. If a 2 −4ac

If D>0 , the parabola cuts the x-axis at exactly two distinct points. The roots are distinct. This case is shown in the above figure in a, where the quadratic polynomial cuts the x-axis at two distinct points.

If D=0 , the parabola just touches the x-axis at one point and the rest of the parabola lies above or below the x-axis. In this case, the roots are equal. This case is shown in the above figure in b, where the quadratic polynomial touches the x-axis at only one point .

If D<0 , the parabola lies entirely above or below the x-axis and there is no point of contact with the x-axis. In this case, there are no real roots. This case is shown in the above figure in c, where the quadratic polynomial neither cuts nor touches the x-axis.

Formation of a quadratic equation from its roots

To find out the standard form of a quadratic equation when the roots are given:

Let α and β be the roots of the quadratic equation ax 2 +bx+c=0. Then,

(x−α)(x−β)=0

On expanding, we get,

x 2 −(α+β)x+αβ=0, which is the standard form of the quadratic equation.

Here, a=1,b=−(α+β) and c=αβ.

Example: Form the quadratic equation if the roots are −3 and 4.

Solution: Given -3 and 4 are the roots of the equation.

Sum of roots = -3 + 4 = 1

Product of the roots = (-3).(4) = -12

As we know, the standard form of a quadratic equation is:

x 2 − (sum of roots)x + (product of roots) = 0

Therefore, by putting the values, we get

x 2 – x – 12 = 0

Which is the required quadratic equation.

Sum and Product of Roots of a Quadratic Equation

Sum of roots = α + β =-b/a

Product of roots = αβ = c/a

Example: Given, x 2 − 5x + 8 = 0 is the quadratic equation. Find the sum and product of its roots.

Solution: x 2 − 5x + 8 = 0 is the quadratic equation given in the form of ax 2 + bx + c = 0. Hence, a = 1 b = -5 c = 8

Sum of roots = -b/a = 5

Product of roots = c/a = 8

To know more about Sum and Product of Roots of a Quadratic equation, visit here .

For more information on Sum and Product of Roots of a Quadratic equation, watch the below video

Related Links:

• NCERT Solutions for Class 10 Maths Chapter 4 Quadratic Equations
• NCERT Exemplar Class 10 Maths Solutions for Chapter 4 – Quadratic Equations
• RD Sharma Solutions for Class 10 Maths Chapter 8 Quadratic Equations
• Class 10 Maths Chapter 4 Quadratic Equations MCQs
• Important Questions Class 10 Maths Chapter 4 Quadratic Equations

Practice Questions on Quadratic Equations Class 10

1. Check whether the following are quadratic equations: (i) (x – 2) 2 + 1 = 2x – 3 (ii) x(x + 1) + 8 = (x + 2) (x – 2) (iii) x (2x + 3) = x 2 + 1 (iv) (x + 2) 3 = x 3 – 4 2. Find two numbers whose sum is 27 and product is 182. 3. Find the roots of 4x 2 + 3x + 5 = 0 by the method of completing the square. 4. Find the roots of the quadratic equation 3x 2 – 5x + 2 = 0, if they exist, using the quadratic formula. 5. Find the values of k for which the quadratic equation kx(x – 2) + 6 = 0 has two equal roots.

Put your understanding of this concept to test by answering a few MCQs. Click ‘Start Quiz’ to begin!

Select the correct answer and click on the “Finish” button Check your score and answers at the end of the quiz

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Nature of Roots

• Cube Root of 8
• Cube Root of 2
• Cube Root of 81
• Cube Root of 64
• Cube Root of 27
• Cube Root of 47
• Cube Root of 343
• Cube Root of 729
• cube root of 216
• Cube Root of 125
• Derivative of Root x
• Cube Root of 1728
• Cube Root of 1000
• Cube Root of 1331
• Root Pressure in Plants
• Square Root of 4
• Fifth root of a number
• N-th root of a number

Roots are the solutions of an equation. The Nature of Roots in mathematics refers to the characteristics and properties of solutions to algebraic equations. These roots represent the values that make the equation true. Understanding the nature of roots is essential for solving equations in science and engineering to analyzing data in statistics. Depending on the equation, roots can be real or complex, and their behavior can provide insights into mathematical relationships. Our context of root in this article is for Quadratic Equations . Nature of Roots is important for Class 10 students.

In this article, we will learn about what are the roots of a quadratic equation, how to determine the nature of roots of a quadratic equation specifying different cases, and solve examples based on the nature of roots.

Table of Content

What are the Roots of Quadratic Equation?

Nature of roots of quadratic equation, different cases of nature of roots, nature of roots – summary.

• Solved Examples

In the context of quadratic equations , the term “roots” refers to the values of the variable (usually denoted as “x”) that satisfy the equation, making it true. We know that the standard representation of a Quadratic Equation is given as ax 2 + bx + c = 0. The roots of a quadratic equation are the values of “x” that, when substituted into the equation, make the equation true (i.e., equal to zero). There can be zero, one, or two real roots (values of “x”) depending on the discriminant (the value inside the square root) of the equation.

The roots of a Quadratic Equations is calculated using Quadratic Formula given below:

x = (-b ± √D)/2a Where, b is coffecicent of x, D is Discriminant, and a is coefficient of x 2 .

In the above formula it is the Value of Discriminant that determines the nature of roots of a quadratic equation. The details of the Nature of Roots depending upon the value of discriminant of a quadratic equation has been discussed below.

Read more about Roots of Quadratic Equation .

This is a concept discussed in mathematics, especially when dealing with quadratic equations. The nature of the roots of a quadratic equation describes the characteristics of the “solutions” which are also known as the “roots” of that Quadratic equation. Quadratic equations are typically in the form:

Discriminant Formula

The nature of the roots for a quadratic equation given as ax 2 + bx + c is determined by the discriminant (D), which is calculated as:

D = b 2 – 4ac

Based on the value of the Discriminant (D) , you can determine the nature of the roots as follows.

The value of Discriminant obtained is used to calculate the roots of a quadratic equation which is done by using quadratic formula given as

x = (-b±√D)/2a

Learn more about Discriminant Formulas for Quadratic Equations .

The nature of roots depends on the value of the Discriminant obtained for a given quadratic equation. Hence, the different cases of the nature of roots has been listed below:

• D is Perfect Square
• D is not Perfect Square

These conditions for nature of roots have been discussed extensively in the article below:

D > 0 (Positive Discriminant)

• Here the discrimination will be positive.

D = 0 (Zero Discriminant)

• Here the discriminant will be equal to zero.

D < 0 (Negative Discriminant)

• Here the discriminant will be negative.

D is a Perfect Square

• Example: If D = 25, which is 5 2 , it’s a perfect square discriminant. The equation has real roots: x = (-b ± 5) / (2a).

D is not a Perfect Square

• Example: For D = 8, which is not a perfect square, the equation has two distinct irrational roots: x = (-b ± √8) / (2a)

The whole concept of Nature of Roots discussed in the article has been summarized below:

Understanding the nature of roots is essential in various fields of mathematics and science, including algebra, calculus, and physics, as it helps determine the behavior and characteristics of solutions to quadratic equations.

Also, Check

Algebra Algebraic Expressions Solving Cubic Equations

Nature of Roots Solved Examples

Example 1. Find the discriminant of the quadratic equation 2x 2 – 3x + 1 = 0.

Given is a Quadratic equation In the given equation, a = 2, b = -3, and c = 1. D = (-3)² – 4(2)(1) ⇒ D = 9 – 8 ⇒ D = 1 So, the discriminant is D = 1. As the discriminant is 1 ( Which is greater than 0), The Equation will have 2 distinct real roots.

Example 2. Find the discriminant of the quadratic equation x 2 + 4x + 4 = 0.

In this equation, a = 1, b = 4, and c = 4. D = (4)² – 4(1)(4) ⇒ D = 16 – 16 ⇒ D = 0 So, the discriminant is D = 0. As the discriminant is equal to 0, the same real roots The roots for the above Quadratic equation are 2,2

Example 3. Find the discriminant of the quadratic equation 3x² – 6x + 9 = 0.

In this equation, a = 3, b = -6, and c = 9. D = (-6)² – 4(3)(9) ⇒ D = 36 – 108 ⇒ D = -72 So, the discriminant is D = -72. As the discriminant is negative (<0) the equation will have the roots both roots are complex and will be conjugate pairs.

Example 4. Find the nature of roots for the Equation: x 2 – 4x + 4 = 0

In this equation x 2 – 4x + 4 = 0 a=1 , b=-4 and c=4. Discriminant (D) = b 2 – 4ac = (-4) 2 – 4(1)(4) = 0 Since D = 0, the roots are real and equal.

Example 5. Find the nature of the roots for the Equation: x 2 + 6x + 9 = 0

In this equation x 2 + 6x + 9 = 0 a=1 , b=6 and c=9 Discriminant (D) = b 2 – 4ac = (6) 2 – 4(1)(9) = 36 – 36 = 0 Since D = 0, the roots are real and equal, but they are -3, a repeated root. Roots = -3,-3.

Example 6. Find the nature of roots for the Equation: 3x 2 – 2x + 1 = 0

In this equation 3x 2 – 2x + 1 = 0 a=3 , b=-2 and c=1 Discriminant (D) = b 2 – 4ac = (-2) 2 – 4(3)(1) = 4 – 12 = -8 Since D < 0, the roots are complex.

Nature of Roots – Practice Questions

Q1. Determine the nature of roots for the equation 2x 2 – 5x + 2 = 0.

Q2. Find the nature of roots for the equation 4x 2 + 12x + 9 = 0.

Q3. What is the nature of roots for the equation 3x 2 – 7x + 4 = 0?

Q4. Determine the nature of roots for the equation x 2 + 6x + 9 = 0.

Q5. Find the nature of roots for the equation 6x 2 – 11x + 4 = 0.

Nature of Roots – FAQs

What is the nature of roots.

The nature of roots is the nature of solutions of a quadratic equation. Based on nature of roots, they can be real roots, complex roots, equal roots, and imaginary roots.

What is the Nature of the Roots Formula?

The nature of roots is described with the discriminant of the equation. The Discriminant formula, D = b 2 – 4ac, determines the nature of roots in a quadratic equation. If D > 0, there are two distinct real roots; if D = 0, there’s one real root (equal roots); and if D < 0, there are no real roots, only complex roots.

How to Find the Nature of Roots?

To find the nature of roots of a quadratic equation ax 2 + bx + c = 0, Calculate the discriminant (D) using the formula: D = b 2 – 4ac. Analyze the value of the discriminant: If D > 0, the equation has two distinct real roots. If D = 0, the equation has one real root (equal roots). If D < 0, the equation has no real roots, only complex roots.

What if the Discriminant is not a Perfect Square?

If Discriminat is not a perfect square then roots are irrational and distinct

Can a Quadratic Equation have more than Two Real Roots?

No, a quadratic equation can have at most two real roots. This is a fundamental property of quadratic equations.

How can I use the Nature of Roots to solve Real-World Problems?

The nature of roots can help you make informed decisions in various fields, such as physics, engineering, economics, and computer science. It aids in understanding the behavior of systems and finding solutions to problems modeled by quadratic equations.

What if the Discriminant is a Perfect Square?

If the discriminant is a perfect square then roots are rational and distinct

What if the Coefficient of ‘a’ in a Quadratic Equation is Zero?

If ‘a’ is zero, it’s not a quadratic equation but a linear equation, and it will have a single root.

9. Can you have Complex Roots with a Positive Discriminant?

No, complex roots are associated with a negative discriminant. A positive discriminant implies two distinct real sources.

10. What we have study in Nature of Roots Class 10?

in Nature of Roots Class 10 we have tom learn the conditions for the various nature of roots. The nature of roots for class 10 has been discussed below: D > 0: Two distinct real roots. D = 0: One real root (repeated). D < 0: Two complex (conjugate) roots. D is Perfect Square: Rational & Distinct Roots D is not a Perfect Square: Irrational & Distinct Roots Also see, Quadratic Equation Class 10 Notes.

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Molecular Study on the DKP Equation in (1 + 3) Dimensions with Isotropic Oscillator plus Inverse Quadratic Potential in Non-Commutative Space

• Published: 13 May 2024
• Volume 63 , article number  123 , ( 2024 )

Cite this article

• O. J. Oluwadare 1 ,
• T. O. Abiola 1 ,
• E. A. Odo 1 ,
• O. Olubosede 1 &
• K. J. Oyewumi 2  

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The non-commutative DKP equation in (1 + 3) dimension with isotropic oscillator plus inverse quadratic potential are solved within the Nikiforov-Uvarov method. The energy eigenvalues equation and the eigenfunctions are obtained. The energy shift arises as a result of space non-commutativity is evaluated within the ambit of perturbation theory. Using the molecular constants for some molecules, the effect of non-commutative space on the behaviour of some molecules are studied. The results revealed that the space noncommutativity modify the bound state energy levels significantly for all the molecules as it was seen that the energy shifts increases for any increase in perturbation parameter \(\uptheta\) .

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Dong, S.H.: Wave Equation in Higher Dimensions, p. 136. Springer Netherland, Dordrecht (2011)

Book   Google Scholar  

Ikhdair, S.M., Falaye, B.J., Hamzavi, M.: Approximate eigensolutions of the deformed Woods-Saxon potential via AIM. Chin. Phys. Lett. 30 , 020305 (2012)

Article   Google Scholar  

Ikhdair, S.M., Falaye, B.J.: Approximate analytical solutions to relativistic and non-relativistic Pӧschl-Teller potential with its Thermodynamic properties. Chem. Phys. 421 , 84–95 (2013)

Oyewumi, K.J., Oluwadare, O.J., Sen, K.D., Babalola, O.A.: Bound state solutions of the Deng-Fan molecular potential with the Pekeris-type approximation using the Nikiforov-Uvarov (N–U) method. J. Math. Chem. 51 , 976 (2013)

Article   MathSciNet   Google Scholar  

Oyewumi, K.J., Falaye, B.J., Onate, C.A., Oluwadare, O.J., Yahya, W.A.: Thermodynamic properties and approximate solutions of Schrödinger equation with the shifted Deng-fan potential model. Mol. Phys. 112 , 127–141 (2013)

Article   ADS   Google Scholar  

Oluwadare, O.J., Oyewumi, K.J.: Non-relativistic treatment of a generalized inverse quadratic Yukawa potential. Chin. Phys. Lett. 34 , 110301 (2017)

Oluwadare, O.J., Oyewumi, K.J.: Energy spectra and the expectation values of diatomic molecules confined by the shifted Deng-Fan potential. Eur. Phys. J. Plus 133 , 422 (2018)

Onate, C.A., Okon, I.B., Onyeaju, M.C., Antia, A.D.: Approximate Solutions of the Schrodinger Equation for a ¨ Momentum-Dependent potential. J. Nig. Soc. Phys. Sci. 4 , 242–250 (2022)

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Khordad, R., Ghanbari, A.: Prediction of Thermodynamic functions of TiC: Morse Ring-Shaped Potential. J. Low Temp. Phys. 199 , 1–13 (2020)

Servatkhah, M., Khordad, R., Ghanbari, A.: Accurate prediction of Thermodynamic functions of H 2 and LiH using theoretical calculations. Intl J. Thermophys. 41 , 37 (2020)

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Onyeaju, M.C., Onate, C.A.: Vibrational entropy and complexity measures in modified Pöschl-Teller Plus Woods-Saxon potential. Few-Body Syst. 61 , 21 (2020)

Lǚtfǚoğlu, B.C., Ikot, A.N., Okorie, U.S., Ngiangia, A.T.: A Statistical Mechanical Analysis on the Bound State Solution of an Energy-Dependent Deformed Hulthén Potential Energy. Commun. Theor. Phys. 71 , 1127–1138 (2019)

Article   ADS   MathSciNet   Google Scholar  

Oluwadare, O.J., Oyewumi, K.J., Abiola, T.O.: Thermodynamic properties of some diatomic molecules confined by an harmonic oscillating system. Indian J. Phys. 96 , 1921–1928 (2021)

Khordad, R.: Hydrogenic impurity state in a spherical quantum antidote: spin-orbit interaction, relativistic correction, and diamagnetic susceptibility. Eur. Phys. J. B. 85 , 353 (2012)

Oluwadare, O.J., Ilesanmi, E.O., Abiola, T.O., Olubosede, O., Odo, E.A., Ajibade, S.O., Oyewumi, K.J.: Investigating some diatomic molecules bounded by the two-dimensional isotropic oscillator plus inverse quadratic potential in an external magnetic field. Adv. High Energy Phys. 2022 , 6565048 (2022)

Zaim, S., Delenda, Y.: Noncommutative of space-time and the relativistic hydrogen atom. J Phys. Conf. Series 435 , 012020 (2013)

Zaim, S., Khodja, L., Delenda, Y.: Second-order corrections to the noncommutative Klein-Gordon equation. Int. J. Mod. Phys. A 26 , 4133 (2011)

Darrodi, M., Mehraban, H., Hassanabadi, S.: The Klein-Gordon equation with Kratzer potential in the noncommutative space. Mod. Phys. Lett. A 33 , 1850203 (2018)

Saidi, A., Sedra, M.B.: Spin-one (1+3)- dimension DKP equation with modified Kratzer potential in the non-commutative space. Mod. Phys. Lett. A 35 , 2050014 (2020)

Hassanabadi, H., Yazarloo, B.H., Zarrinkamar, S., Rajabi, A.A.: Duffin-Kemmer-Petiau equation under a scalar Coulomb interaction. Phys. Rev. C. 84 , 064003 (2011)

Hassanabadi, H., Kamali, M., Yazarloo, B.H.: Spin-one Duffin-Kemmer-Petiau equation in the presence of Manning-Rosen potential plus a ring-shaped-like potential. Can. J. Phys. 92 , 1–7 (2014)

Zarrinkamar, S., Forouhandeh, S.F., Yazarloo, B.H., Hassanabadi, H.: scattering states of the Duffin-Kemmer-Petiau equation for the Hulthén potential. Eur. Phys. J. Plus 128 , 109 (2013)

Salehi, N., Hassanabadi, H.: Scattering amplitude of DKP equation for the Yukawa potential for J=0. Eur. Phys. J. A 51 , 100 (2015)

Oluwadare, O.J., Oyewumi, K.J.: Scattering state solutions of the Duffin-Kemmer-Petiau equation with the Varshni potential model. Eur. Phys. J. A 53 , 29 (2017)

Oluwadare, O.J., Oyewumi, K.J.: Approximate Scattering State Solutions of DKPE and SSE with Hellmann Potential. Adv. High Energy Phys. 2018 , 521404 (2018)

Khordad, R., Mohammadi, S.A.: Analytical study of absorption threshold frequency of tuned quantum dot/ring system. Phys. B Condensed Matt. 647 , 414381 (2022)

Khordad, R., Mohammadi, S.A.: Simultaneous effects of pressure, temperature and external magnetic field on absorption threshold frequency of tuned quantum dot/ring systems: an analytical study. J. Comput. Elect. 22 , 641–647 (2023)

Nikiforov, A.N., Uvarov, V.B.: Special functions of Mathematical Physics. Birkhauser, Berlin (1988)

Rahimov, H., Nikoofard, H., Zarrinkamar, S., Hassanabadi, H.: Any l- state solutions of the Schrödinger equation for the Modified Woods-Saxon potential in arbitrary dimensions. Appl. Math. Comp. 219 , 4710–4717 (2013)

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Oluwadare, O.J., Abiola, T.O., Odo, E.A. et al. Molecular Study on the DKP Equation in (1 + 3) Dimensions with Isotropic Oscillator plus Inverse Quadratic Potential in Non-Commutative Space. Int J Theor Phys 63 , 123 (2024). https://doi.org/10.1007/s10773-024-05662-3

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Search for new biologically active compounds: in vitro studies of antitumor and antimicrobial activity of dirhodium( ii , ii ) paddlewheel complexes †.

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a University of Kragujevac, Faculty of Medical Sciences, Department of Medical Biochemistry, Svetozara Markovića 69, 34000 Kragujevac, Serbia

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c University of Kragujevac, Faculty of Medical Sciences, Department of Pharmacy, Svetozara Markovića 69, 34000 Kragujevac, Serbia

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Four neutral Rh1–Rh4 complexes of the general formula [Rh 2 (CH 3 COO) 4 L 2 ], where L is an N -alkylimidazole ligand, were synthesized and characterized using various spectroscopic techniques, and in the case of Rh4 the crystal structure was confirmed. Investigation of the interactions of these complexes with HSA by fluorescence spectroscopy revealed that the binding constants K b are moderately strong (∼10 4 M −1 ), and site-marker competition experiments showed that the complexes bind to Heme site III (subdomain IB). Competitive binding studies for CT DNA using EB and HOE showed that the complexes bind to the minor groove, which was also confirmed by viscosity experiments. Molecular docking confirmed the experimental data for HSA and CT DNA. Antimicrobial tests showed that the Rh2–Rh4 complexes exerted a strong inhibitory effect on G + bacteria B. cereus and G − bacteria V. parahaemolyticus as well as on the yeast C. tropicalis , which showed a higher sensitivity compared to fluconazole. The cytotoxic activity of Rh1–Rh4 complexes tested on three cancer cell lines (HeLa, HCT116 and MDA-MB-231) and on healthy MRC-5 cells showed that all investigated complexes elicited more efficient cytotoxicity on all tested tumor cells than on control cells. Investigation of the mechanism of action revealed that the Rh1–Rh4 complexes inhibit cell proliferation via different mechanisms of action, namely apoptosis (increase in expression of the pro-apoptotic Bax protein and caspase-3 protein in HeLa and HCT116 cells; changes in mitochondrial potential and mitochondrial damage; release of cytochrome c from the mitochondria; cell cycle arrest in G2/M phase in both HeLa and HCT116 cells together with a decrease in the expression of cyclin A and cyclin B) and autophagy (reduction in the expression of the protein p62 in HeLa and HCT116 cells).

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Search for new biologically active compounds: in vitro studies of antitumor and antimicrobial activity of dirhodium( II , II ) paddlewheel complexes

M. Mitrović, M. B. Djukić, M. Vukić, I. Nikolić, M. D. Radovanović, J. Luković, I. P. Filipović, S. Matić, T. Marković, O. R. Klisurić, S. Popović, Z. D. Matović and M. S. Ristić, Dalton Trans. , 2024, Advance Article , DOI: 10.1039/D4DT01082E

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Does a perceptual gap lead to actions against digital misinformation? A third-person effect study among medical students

• Zongya Li   ORCID: orcid.org/0000-0002-4479-5971 1 &
• Jun Yan   ORCID: orcid.org/0000-0002-9539-8466 1  

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We are making progress in the fight against health-related misinformation, but mass participation and active engagement are far from adequate. Focusing on pre-professional medical students with above-average medical knowledge, our study examined whether and how third-person perceptions (TPP), which hypothesize that people tend to perceive media messages as having a greater effect on others than on themselves, would motivate their actions against misinformation.

We collected the cross-sectional data through a self-administered paper-and-pencil survey of 1,500 medical students in China during April 2022.

Structural equation modeling (SEM) analysis, showed that TPP was negatively associated with medical students’ actions against digital misinformation, including rebuttal of misinformation and promotion of corrective information. However, self-efficacy and collectivism served as positive predictors of both actions. Additionally, we found professional identification failed to play a significant role in influencing TPP, while digital misinformation self-efficacy was found to broaden the third-person perceptual gap and collectivism tended to reduce the perceptual bias significantly.

Conclusions

Our study contributes both to theory and practice. It extends the third-person effect theory by moving beyond the examination of restrictive actions and toward the exploration of corrective and promotional actions in the context of misinformation., It also lends a new perspective to the current efforts to counter digital misinformation; involving pre-professionals (in this case, medical students) in the fight.

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Introduction

The widespread persistence of misinformation in the social media environment calls for effective strategies to mitigate the threat to our society [ 1 ]. Misinformation has received substantial scholarly attention in recent years [ 2 ], and solution-oriented explorations have long been a focus but the subject remains underexplored [ 3 ].

Health professionals, particularly physicians and nurses, are highly expected to play a role in the fight against misinformation as they serve as the most trusted information sources regarding medical topics [ 4 ]. However, some barriers, such as limitations regarding time and digital skills, greatly hinder their efforts to tackle misinformation on social media [ 5 ].

Medical students (i.e., college students majoring in health/medical science), in contrast to medical faculty, have a greater potential to become the major force in dealing with digital misinformation as they are not only equipped with basic medical knowledge but generally possess greater social media skills than the former generation [ 6 ]. Few studies, to our knowledge, have tried to explore the potential of these pre-professionals in tackling misinformation. Our research thus fills the gap by specifically exploring how these pre-professionals can be motivated to fight against digital health-related misinformation.

The third-person perception (TPP), which states that people tend to perceive media messages as having a greater effect on others than on themselves [ 7 ], has been found to play an important role in influencing individuals’ coping strategies related to misinformation. But empirical exploration from this line of studies has yielded contradictory results. Some studies revealed that individuals who perceived a greater negative influence of misinformation on others than on themselves were more likely to take corrective actions to debunk misinformation [ 8 ]. In contrast, some research found that stronger TPP reduced individuals’ willingness to engage in misinformation correction [ 9 , 10 ]. Such conflicting findings impel us to examine the association between the third-person perception and medical students’ corrective actions in response to misinformation, thus attempting to unveil the underlying mechanisms that promote or inhibit these pre-professionals’ engagement with misinformation.

Researchers have also identified several perceptual factors that motivate individuals’ actions against misinformation, especially efficacy-related concepts (e.g., self-efficacy and health literacy) and normative variables (e.g., subjective norms and perceived responsibility) [ 3 , 8 , 9 ]. However, most studies devote attention to the general population; little is known about whether and how these factors affect medical students’ intentions to deal with misinformation. We recruited Chinese medical students in order to study a social group that is mutually influenced by cultural norms (collectivism in Chinese society) and professional norms. Meanwhile, systematic education and training equip medical students with abundant clinical knowledge and good levels of eHealth literacy [ 5 ], which enable them to have potential efficacy in tackling misinformation. Our study thus aims to examine how medical students’ self-efficacy, cultural norms (i.e., collectivism) and professional norms (i.e., professional identification) impact their actions against misinformation.

Previous research has found self-efficacy to be a reliable moderator of optimistic bias, the tendency for individuals to consider themselves as less likely to experience negative events but more likely to experience positive events as compared to others [ 11 , 12 , 13 ]. As TPP is thought to be a product of optimistic bias, accordingly, self-efficacy should have the potential to influence the magnitude of third-person perception [ 14 , 15 ]. Meanwhile, scholars also suggest that the magnitude of TPP is influenced by social distance corollary [ 16 , 17 ]. Simply put, individuals tend to perceive those who are more socially distant from them to be more susceptible to the influence of undesirable media than those who are socially proximal [ 18 , 19 , 20 ]. From a social identity perspective, collectivism and professional identification might moderate the relative distance between oneself and others while the directions of such effects differ [ 21 , 22 ]. For example, collectivists tend to perceive a smaller social distance between self and others as “they are less likely to view themselves as distinct or unique from others” [ 23 ]. In contrast, individuals who are highly identified with their professional community (i.e., medical community) are more likely to perceive a larger social distance between in-group members (including themselves) and out-group members [ 24 ]. In this way, collectivism and professional identification might exert different effects on TPP. On this basis, this study aims to examine whether and how medical students’ perceptions of professional identity, self-efficacy and collectivism influence the magnitude of TPP and in turn influence their actions against misinformation.

Our study builds a model that reflects the theoretical linkages among self-efficacy, collectivism, professional identity, TPP, and actions against misinformation. The model, which clarifies the key antecedents of TPP and examines the mediating role of TPP, contribute to the third-person effect literature and offer practical contributions to countering digital misinformation.

Context of the study

As pre-professionals equipped with specialized knowledge and skills, medical students have been involved in efforts in health communication and promotion during the pandemic. For instance, thousands of medical students have participated in various volunteering activities in the fight against COVID-19, such as case data visualization [ 25 ], psychological counseling [ 26 ], and providing online consultations [ 27 ]. Due to the shortage of medical personnel and the burden of work, some medical schools also encouraged their students to participate in health care assistance in hospitals during the pandemic [ 28 , 29 ].

The flood of COVID-19 related misinformation has posed an additional threat to and burden on public health. We have an opportunity to address this issue and respond to the general public’s call for guidance from the medical community about COVID-19 by engaging medical students as a main force in the fight against coronavirus related misinformation.

Literature review

The third-person effect in the misinformation context.

Originally proposed by Davison [ 7 ], the third-person effect hypothesizes that people tend to perceive a greater effect of mass media on others than on themselves. Specifically, the TPE consists of two key components: the perceptual and the behavioral [ 16 ]. The perceptual component centers on the perceptual gap where individuals tend to perceive that others are more influenced by media messages than themselves. The behavioral component refers to the behavioral outcomes of the self-other perceptual gap in which people act in accordance with such perceptual asymmetry.

According to Perloff [ 30 ], the TPE is contingent upon situations. For instance, one general finding suggests that when media messages are considered socially undesirable, nonbeneficial, or involving risks, the TPE will get amplified [ 16 ]. Misinformation characterized as inaccurate, misleading, and even false, is regarded as undesirable in nature [ 31 ]. Based on this line of reasoning, we anticipate that people will tend to perceive that others would be more influenced by misinformation than themselves.

Recent studies also provide empirical evidence of the TPE in the context of misinformation [ 32 ]. For instance, an online survey of 511 Chinese respondents conducted by Liu and Huang [ 33 ] revealed that individuals would perceive others to be more vulnerable to the negative influence of COVID-19 digital disinformation. An examination of the TPE within a pre-professional group – the medical students–will allow our study to examine the TPE scholarship in a particular population in the context of tackling misinformation.

Why TPE occurs among medical students: a social identity perspective

Of the works that have provided explanations for the TPE, the well-known ones include self-enhancement [ 34 ], attributional bias [ 35 ], self-categorization theory [ 36 ], and the exposure hypothesis [ 19 ]. In this study, we argue for a social identity perspective as being an important explanation for third-person effects of misinformation among medical students [ 36 , 37 ].

The social identity explanation suggests that people define themselves in terms of their group memberships and seek to maintain a positive self-image through favoring the members of their own groups over members of an outgroup, which is also known as downward comparison [ 38 , 39 ]. In intergroup settings, the tendency to evaluate their ingroups more positively than the outgroups will lead to an ingroup bias [ 40 ]. Such an ingroup bias is typically described as a trigger for the third-person effect as individuals consider themselves and their group members superior and less vulnerable to undesirable media messages than are others and outgroup members [ 20 ].

In the context of our study, medical students highly identified with the medical community tend to maintain a positive social identity through an intergroup comparison that favors the ingroup and derogates the outgroup (i.e., the general public). It is likely that medical students consider themselves belonging to the medical community and thus are more knowledgeable and smarter than the general public in health-related topics, leading them to perceive the general public as more vulnerable to health-related misinformation than themselves. Accordingly, we propose the following hypothesis:

H1: As medical students’ identification with the medical community increases, the TPP concerning digital misinformation will become larger.

What influences the magnitude of TPP

Previous studies have demonstrated that the magnitude of the third-person perception is influenced by a host of factors including efficacy beliefs [ 3 ] and cultural differences in self-construal [ 22 , 23 ]. Self-construal is defined as “a constellation of thoughts, feelings, and actions concerning the relationship of the self to others, and the self as distinct from others” [ 41 ]. Markus and Kitayama (1991) identified two dimensions of self-construal: Independent and interdependent. Generally, collectivists hold an interdependent view of the self that emphasizes harmony, relatedness, and places importance on belonging, whereas individualists tend to have an independent view of the self and thus view themselves as distinct and unique from others [ 42 ]. Accordingly, cultural values such as collectivism-individualism should also play a role in shaping third-person perception due to the adjustment that people make of the self-other social identity distance [ 22 ].

Set in a Chinese context aiming to explore the potential of individual-level approaches to deal with misinformation, this study examines whether collectivism (the prevailing cultural value in China) and self-efficacy (an important determinant of ones’ behavioral intentions) would affect the magnitude of TPP concerning misinformation and how such impact in turn would influence their actions against misinformation.

The impact of self-efficacy on TPP

Bandura [ 43 ] refers to self-efficacy as one’s perceived capability to perform a desired action required to overcome barriers or manage challenging situations. He also suggests understanding self-efficacy as “a differentiated set of self-beliefs linked to distinct realms of functioning” [ 44 ]. That is to say, self-efficacy should be specifically conceptualized and operationalized in accordance with specific contexts, activities, and tasks [ 45 ]. In the context of digital misinformation, this study defines self-efficacy as one’s belief in his/her abilities to identify and verify misinformation within an affordance-bounded social media environment [ 3 ].

Previous studies have found self-efficacy to be a reliable moderator of biased optimism, which indicates that the more efficacious individuals consider themselves, the greater biased optimism will be invoked [ 12 , 23 , 46 ]. Even if self-efficacy deals only with one’s assessment of self in performing a task, it can still create the other-self perceptual gap; individuals who perceive a higher self-efficacy tend to believe that they are more capable of controlling a stressful or challenging situation [ 12 , 14 ]. As such, they are likely to consider themselves less vulnerable to negative events than are others [ 23 ]. That is, individuals with higher levels of self-efficacy tend to underestimate the impact of harmful messages on themselves, thereby widening the other-self perceptual gap.

In the context of fake news, which is closely related to misinformation, scholars have confirmed that fake news efficacy (i.e., a belief in one’s capability to evaluate fake news [ 3 ]) may lead to a larger third-person perception. Based upon previous research evidence, we thus propose the following hypothesis:

H2: As medical students’ digital misinformation self-efficacy increases, the TPP concerning digital misinformation will become larger.

The influence of collectivism on TPP

Originally conceptualized as a societal-level construct [ 47 ], collectivism reflects a culture that highlights the importance of collective goals over individual goals, defines the self in relation to the group, and places great emphasis on conformity, harmony and interdependence [ 48 ]. Some scholars propose to also examine cultural values at the individual level as culture is embedded within every individual and could vary significantly among individuals, further exerting effects on their perceptions, attitudes, and behaviors [ 49 ]. Corresponding to the construct at the macro-cultural level, micro-psychometric collectivism which reflects personality tendencies is characterized by an interdependent view of the self, a strong sense of other-orientation, and a great concern for the public good [ 50 ].

A few prior studies have indicated that collectivism might influence the magnitude of TPP. For instance, Lee and Tamborini [ 23 ] found that collectivism had a significant negative effect on the magnitude of TPP concerning Internet pornography. Such an impact can be understood in terms of biased optimism and social distance. Collectivists tend to view themselves as an integral part of a greater social whole and consider themselves less differentiated from others [ 51 ]. Collectivism thus would mitigate the third-person perception due to a smaller perceived social distance between individuals and other social members and a lower level of comparative optimism [ 22 , 23 ]. Based on this line of reasoning, we thus propose the following hypothesis:

H3: As medical students’ collectivism increases, the TPP concerning digital misinformation will become smaller.

Behavioral consequences of TPE in the misinformation context

The behavioral consequences trigged by TPE have been classified into three categories: restrictive actions refer to support for censorship or regulation of socially undesirable content such as pornography or violence on television [ 52 ]; corrective action is a specific type of behavior where people seek to voice their own opinions and correct the perceived harmful or ambiguous messages [ 53 ]; promotional actions target at media content with desirable influence, such as advocating for public service announcements [ 24 ]. In a word, restriction, correction and promotion are potential behavioral outcomes of TPE concerning messages with varying valence of social desirability [ 16 ].

Restrictive action as an outcome of third-person perceptual bias (i.e., the perceptual component of TPE positing that people tend to perceive media messages to have a greater impact on others than on themselves) has received substantial scholarly attention in past decades; scholars thus suggest that TPE scholarship to go beyond this tradition and move toward the exploration of corrective and promotional behaviors [ 16 , 24 ]. Moreover, individual-level corrective and promotional actions deserve more investigation specifically in the context of countering misinformation, as efforts from networked citizens have been documented as an important supplement beyond institutional regulations (e.g., drafting policy initiatives to counter misinformation) and platform-based measures (e.g., improving platform algorithms for detecting misinformation) [ 8 ].

In this study, corrective action specifically refers to individuals’ reactive behaviors that seek to rectify misinformation; these include such actions as debunking online misinformation by commenting, flagging, or reporting it [ 3 , 54 ]. Promotional action involves advancing correct information online, including in response to misinformation that has already been disseminated to the public [ 55 ].

The impact of TPP on corrective and promotional actions

Either paternalism theory [ 56 ] or the protective motivation theory [ 57 ] can act as an explanatory framework for behavioral outcomes triggered by third-person perception. According to these theories, people act upon TPP as they think themselves to know better and feel obligated to protect those who are more vulnerable to negative media influence [ 58 ]. That is, corrective and promotional actions as behavioral consequences of TPP might be driven by a protective concern for others and a positive sense of themselves.

To date, several empirical studies across contexts have examined the link between TPP and corrective actions. Koo et al. [ 8 ], for instance, found TPP was not only positively related to respondents’ willingness to correct misinformation propagated by others, but also was positively associated with their self-correction. Other studies suggest that TPP motivates individuals to engage in both online and offline corrective political participation [ 59 ], give a thumbs down to a biased story [ 60 ], and implement corrective behaviors concerning “problematic” TV reality shows [ 16 ]. Based on previous research evidence, we thus propose the following hypothesis:

H4: Medical students with higher degrees of TPP will report greater intentions to correct digital misinformation.

Compared to correction, promotional behavior has received less attention in the TPE research. Promotion commonly occurs in a situation where harmful messages have already been disseminated to the public and others appear to have been influenced by these messages, and it serves as a remedial action to amplify messages with positive influence which may in turn mitigate the detrimental effects of harmful messages [ 16 ].

Within this line of studies, however, empirical studies provide mixed findings. Wei and Golan [ 24 ] found a positive association between TPP of desirable political ads and promotional social media activism such as posting or linking the ad on their social media accounts. Sun et al. [ 16 ] found a negative association between TPP regarding clarity and community-connection public service announcements (PSAs) and promotion behaviors such as advocating for airing more PSAs in TV shows.

As promotional action is still underexplored in the TPE research, and existing evidence for the link between TPP and promotion is indeed mixed, we thus propose an exploratory research question:

RQ1: What is the relationship between TPP and medical students’ intentions to promote corrective information?

The impact of self-efficacy and collectivism on actions against misinformation

According to social cognitive theory, people with higher levels of self-efficacy tend to believe they are competent and capable and are more likely to execute specific actions [ 43 ]. Within the context of digital misinformation, individuals might become more willing to engage in misinformation correction if they have enough knowledge and confidence to evaluate information, and possess sufficient skills to verify information through digital tools and services [ 61 ].

Accordingly, we assumed medical students with higher levels of digital misinformation self-efficacy would be likely to become more active in the fight against misinformation.

H5: Medical students with higher levels of digital misinformation self-efficacy will report greater intentions to (a) correct misinformation and (b) promote corrective information on social media.

Social actions of collectivists are strongly guided by prevailing social norms, collective responsibilities, and common interest, goals, and obligations [ 48 ]. Hence, highly collectivistic individuals are more likely to self-sacrifice for group interests and are more oriented toward pro-social behaviors, such as adopting pro-environmental behaviors [ 62 ], sharing knowledge [ 23 ], and providing help for people in need [ 63 ].

Fighting against misinformation is also considered to comprise altruism, especially self-engaged corrective and promotional actions, as such actions are costly to the actor (i.e., taking up time and energy) but could benefit the general public [ 61 ]. Accordingly, we assume collectivism might play a role in prompting people to engage in reactive behaviors against misinformation.

It is also noted that collectivist values are deeply rooted in Chinese society and were especially strongly advocated during the outbreak of COVID-19 with an attempt to motivate prosocial behaviors [ 63 ]. Accordingly, we expected that the more the medical students were oriented toward collectivist values, the more likely they would feel personally obliged and normatively motivated to engage in misinformation correction. However, as empirical evidence was quite limited, we proposed exploratory research questions:

RQ2: Will medical students with higher levels of collectivism report greater intentions to (a) correct misinformation and (b) promote corrective information on social media?

The theoretical model

To integrate both the antecedents and consequences of TPP, we proposed a theoretical model (as shown in Fig. 1 ) to examine how professional identification, self-efficacy and collectivism would influence the magnitude of TPP, and how such impact would in turn influence medical students’ intentions to correct digital misinformation and promote corrective information. Thus, RQ3 was proposed:

RQ3: Will the TPP mediate the impact of self-efficacy and collectivism on medical students’ intentions to (a) correct misinformation, and (b) promote corrective information on social media? Fig. 1 The proposed theoretical model. DMSE = Digital Misinformation Self-efficacy; PIMC = Professional Identification with Medical Community; ICDM = Intention to Correct Digital Misinformation; IPCI = Intention to Promote Corrective Information Full size image

To examine the proposed hypotheses, this study utilized cross-sectional survey data from medical students in Tongji Medical College (TJMC) of China. TJMC is one of the birthplaces of Chinese modern medical education and among the first universities and colleges that offer eight-year curricula on clinical medicine. Further, TJMC is located in Wuhan, the epicenter of the initial COVID-19 outbreaks, thus its students might find the pandemic especially relevant – and threatening – to them.

The survey instrument was pilot tested using a convenience sample of 58 respondents, leading to minor refinements to a few items. Upon approval from the university’s Institutional Research Board (IRB), the formal investigation was launched in TJMC during April 2022. Given the challenges of reaching the whole target population and acquiring an appropriate sampling frame, this study employed purposive and convenience sampling.

We first contacted four school counselors as survey administrators through email with a letter explaining the objective of the study and requesting cooperation. All survey administrators were trained by the principal investigator to help with the data collection in four majors (i.e., basic medicine, clinical medicine, nursing, and public health). Paper-and-pencil questionnaires were distributed to students on regular weekly departmental meetings of each major as students in all grades (including undergraduates, master students, and doctoral students) were required to attend the meeting. The projected time of completion of the survey was approximately 10–15 min. The survey administrators indicated to students that participation was voluntary, their responses would remain confidential and secure, and the data would be used only for academic purposes. Though a total of 1,500 participants took the survey, 17 responses were excluded from the analysis as they failed the attention filters. Ultimately, a total of 1,483 surveys were deemed valid for analysis.

Of the 1,483 respondents, 624 (42.10%) were men and 855 (57.70%) were women, and four did not identify gender. The average age of the sample was 22.00 ( SD  = 2.54, ranging from 17 to 40). Regarding the distribution of respondents’ majors, 387 (26.10%) were in basic medicine, 390 (26.30%) in clinical medicine, 307 (20.70%) in nursing, and 399 (26.90%) in public health. In terms of university class, 1,041 (70.40%) were undergraduates, 291 (19.70%) were working on their master degrees, 146 (9.90%) were doctoral students, and five did not identify their class data.

Measurement of key variables

Perceived effects of digital misinformation on oneself and on others.

Three modified items adapted from previous research [ 33 , 64 ] were employed to measure perceived effects of digital misinformation on oneself. Respondents were asked to indicate to what extent they agreed with the following: (1) I am frequently concerned that the information about COVID-19 I read on social media might be false; (2) Misinformation on social media might misguide my understanding of the coronavirus; (3) Misinformation on social media might influence my decisions regarding COVID-19. The response categories used a 7-point scale, where 1 meant “strongly disagree” and 7 meant “strongly agree.” The measure of perceived effects of digital misinformation on others consisted of four parallel items with the same statement except replacing “I” and “my” with “the general others” and “their”. The three “self” items were averaged to create a measure of “perceived effects on oneself” ( M  = 3.98, SD  = 1.49, α  = 0.87). The three “others” items were also added and averaged to form an index of “perceived effects on others” ( M  = 4.62, SD  = 1.32, α  = 0.87).

The perceived self-other disparity (TPP)

TPP was derived by subtracting perceived effects on oneself from perceived effects on others.

Professional identification with medical community

Professional identification was measured using a three item, 7-point Likert-type scale (1 =  strongly disagree , 7 =  strongly agree ) adapted from previous studies [ 65 , 66 ] by asking respondents to indicate to what extent they agreed with the following statements: (1) I would be proud to be a medical staff member in the future; (2) I am committed to my major; and (3) I will be in an occupation that matches my current major. The three items were thus averaged to create a composite measure of professional identification ( M  = 5.34, SD  = 1.37, α  = 0.88).

Digital misinformation self-efficacy

Modified from previous studies [ 3 ], self-efficacy was measured with three items. Respondents were asked to indicate on a 7-point Linkert scale from 1 (strongly disagree) to 7 (strongly agree) their agreement with the following: (1) I think I can identify misinformation relating to COVID-19 on social media by myself; (2) I know how to verify misinformation regarding COVID-19 by using digital tools such as Tencent Jiaozhen Footnote 1 and Piyao.org.cn Footnote 2 ; (3) I am confident in my ability to identify digital misinformation relating to COVID-19. A composite measure of self-efficacy was constructed by averaging the three items ( M  = 4.38, SD  = 1.14, α  = 0.77).

• Collectivism

Collectivism was measured using four items adapted from previous research [ 67 ], in which respondents were asked to indicate their agreement with the following statements on a 7-point scale, from 1 (strongly disagree) to 7 (strongly agree): (1) Individuals should sacrifice self-interest for the group; (2) Group welfare is more important than individual rewards; (3) Group success is more important than individual success; and (4) Group loyalty should be encouraged even if individual goals suffer. Therefore, the average of the four items was used to create a composite index of collectivism ( M  = 4.47, SD  = 1.30, α  = 0.89).

Intention to correct digital misinformation

We used three items adapted from past research [ 68 ] to measure respondents’ intention to correct misinformation on social media. All items were scored on a 7-point scale from 1 (very unlikely) to 7 (very likely): (1) I will post a comment saying that the information is wrong; (2) I will message the person who posts the misinformation to tell him/her the post is wrong; (3) I will track the progress of social media platforms in dealing with the wrong post (i.e., whether it’s deleted or corrected). A composite measure of “intention to correct digital misinformation” was constructed by adding the three items and dividing by three ( M  = 3.39, SD  = 1.43, α  = 0.81).

Intention to promote corrective information

On a 7-point scale ranging from 1 (very unlikely) to 7 (very likely), respondents were asked to indicate their intentions to (1) Retweet the corrective information about coronavirus on my social media account; (2) Share the corrective information about coronavirus with others through Social Networking Services. The two items were averaged to create a composite measure of “intention to promote corrective information” ( M  = 4.60, SD  = 1.68, r  = 0.77).

Control variables

We included gender, age, class (1 = undergraduate degree; 2 = master degree; 3 = doctoral degree), and clinical internship (0 = none; 1 = less than 0.5 year; 2 = 0.5 to 1.5 years; 3 = 1.5 to 3 years; 4 = more than 3 years) as control variables in the analyses. Additionally, coronavirus-related information exposure (i.e., how frequently they were exposed to information about COVID-19 on Weibo, WeChat, and QQ) and misinformation exposure on social media (i.e., how frequently they were exposed to misinformation about COVID-19 on Weibo, WeChat, and QQ) were also assessed as control variables because previous studies [ 69 , 70 ] had found them relevant to misinformation-related behaviors. Descriptive statistics and bivariate correlations between main variables were shown in Table 1 .

Statistical analysis

We ran confirmatory factor analysis (CFA) in Mplus (version 7.4, Muthén & Muthén, 1998) to ensure the construct validity of the scales. To examine the associations between variables and tested our hypotheses, we performed structural equation modeling (SEM). Mplus was chosen over other SEM statistical package mainly because the current data set included some missing data, and the Mplus has its strength in handling missing data using full-information maximum likelihood imputation, which enabled us to include all available data [ 71 , 72 ]. Meanwhile, Mplus also shows great flexibility in modelling when simultaneously handling continuous, categorical, observed, and latent variables in a variety of models. Further, Mplus provides a variety of useful information in a concise manner [ 73 ].

Table 2 shows the model fit information for the measurement and structural models. Five latent variables were specified in the measurement model. To test the measurement model, we examined the values of Cronbach’s alpha, composite reliability (CR), and average variance extracted (AVE) (Table 1 ). Cronbach’s alpha values ranged from 0.77 to 0.89. The CRs, which ranged from 0.78 to 0.91, exceeded the level of 0.70 recommended by Fornell (1982) and thus confirmed the internal consistency. The AVE estimates, which ranged from 0.54 to 0.78, exceeded the 0.50 lower limit recommended by Fornell and Larcker (1981), and thus supported convergent validity. All the square roots of AVE were greater than the off-diagonal correlations in the corresponding rows and columns [ 74 ]. Therefore, discriminant validity was assured. In a word, our measurement model showed sufficient convergence and discriminant validity.

Five model fit indices–the relative chi-square ratio (χ 2 / df ), the comparative fit index (CFI), the Tucker–Lewis index (TLI), the root mean square error of approximation (RMSEA), and the standardized root-mean-square residual (SRMR) were used to assess the model. Specifically, the normed chi-square between 1 and 5 is acceptable [ 75 ]. TLI and CFI over 0.95 are considered acceptable, SRMR value less than 0.08 and RMSEA value less than 0.06 indicate good fit [ 76 ]. Based on these criteria, the model was found to have an acceptable fit to the data.

Figure 2 presents the results of our hypothesized model. H1 was rejected as professional identification failed to predict TPP ( β  = 0.06, p  > 0.05). Self-efficacy was positively associated with TPP ( β  = 0.14, p  < 0.001) while collectivism was negatively related to TPP ( β  = -0.10, p  < 0.01), lending support to H2 and H3.

Note. N  = 1,483. The coefficients of relationships between latent variables are standardized beta coefficients. Significant paths are indicated by solid line; non-significant paths are indicated by dotted lines. * p  < .05, ** p  < .01; *** p  < .001. DMSE = Digital Misinformation Self-efficacy; PIMC = Professional Identification with Medical Community; ICDM = Intention to Correct Digital Misinformation; IPCI = Intention to Promote Corrective Information

H4 posited that medical students with higher degrees of TPP would report greater intentions to correct digital misinformation. However, we found a negative association between TPP and intentions to correct misinformation ( β  = -0.12, p  < 0.001). H4 was thus rejected. Regarding RQ1, results revealed that TPP was negatively associated with intentions to promote corrective information ( β  = -0.08, p  < 0.05).

Further, our results supported H5 as we found that self-efficacy had a significant positive relationship with corrective intentions ( β  = 0.18, p  < 0.001) and promotional intentions ( β  = 0.32, p  < 0.001). Collectivism was also positively associated with intentions to correct misinformation ( β  = 0.14, p  < 0.001) and promote corrective information ( β  = 0.20, p  < 0.001), which answered RQ2.

Regarding RQ3 (see Table 3 ), TPP significantly mediated the relationship between self-efficacy and intentions to correct misinformation ( β  = -0.016), as well as the relationship between self-efficacy and intentions to promote corrective information ( β  = -0.011). However, TPP failed to mediate either the association between collectivism and corrective intentions ( β  = 0.011, ns ) or the association between collectivism and promotional intentions ( β  = 0.007, ns ).

Recent research has highlighted the role of health professionals and scientists in the fight against misinformation as they are considered knowledgeable, ethical, and reliable [ 5 , 77 ]. This study moved a step further by exploring the great potential of pre-professional medical students to tackle digital misinformation. Drawing on TPE theory, we investigated how medical students perceived the impact of digital misinformation, the influence of professional identification, self-efficacy and collectivism on these perceptions, and how these perceptions would in turn affect their actions against digital misinformation.

In line with prior studies [ 3 , 63 ], this research revealed that self-efficacy and collectivism played a significant role in influencing the magnitude of third-person perception, while professional identification had no significant impact on TPP. As shown in Table 1 , professional identification was positively associated with perceived effects of misinformation on oneself ( r  = 0.14, p  < 0.001) and on others ( r  = 0.20, p  < 0.001) simultaneously, which might result in a diminished TPP. What explains a shared or joint influence of professional identification on self and others? A potential explanation is that even medical staff had poor knowledge about the novel coronavirus during the initial outbreak [ 78 ]. Accordingly, identification with the medical community was insufficient to create an optimistic bias concerning identifying misinformation about COVID-19.

Our findings indicated that TPP was negatively associated with medical students’ intentions to correct misinformation and promote corrective information, which contradicted our hypotheses but was consistent with some previous TPP research conducted in the context of perceived risk [ 10 , 79 , 80 , 81 ]. For instance, Stavrositu and Kim (2014) found that increased TPP regarding cancer risk was negatively associated with behavioral intentions to engage in further cancer information search/exchange, as well as to adopt preventive lifestyle changes. Similarly, Wei et al. (2008) found concerning avian flu news that TPP negatively predicted the likelihood of engaging in actions such as seeking relevant information and getting vaccinated. In contrast, the perceived effects of avian flu news on oneself emerged as a positive predictor of intentions to take protective behavior.

Our study shows a similar pattern as perceived effects of misinformation on oneself were positively associated with intentions to correct misinformation ( r  = 0.06, p  < 0.05) and promote corrective information ( r  = 0.10, p  < 0.001, See Table 1 ). While the reasons for the behavioral patterns are rather elusive, such findings are indicative of human nature. When people perceive misinformation-related risk to be highly personally relevant, they do not take chances. However, when they perceive others to be more vulnerable than themselves, a set of sociopsychological dynamics such as self-defense mechanism, positive illusion, optimistic bias, and social comparison provide a restraint on people’s intention to engage in corrective and promotional actions against misinformation [ 81 ].

In addition to the indirect effects via TPP, our study also revealed that self-efficacy and collectivism serve as direct and powerful drivers of corrective and promotive actions. Consistent with previous literature [ 61 , 68 ], individuals will be more willing to engage in social corrections of misinformation if they possess enough knowledge, skills, abilities, and resources to identify misinformation, as correcting misinformation is difficult and their effort would not necessarily yield positive outcomes. Collectivists are also more likely to engage in misinformation correction as they are concerned for the public good and social benefits, aiming to protect vulnerable people from being misguided by misinformation [ 82 ].

This study offers some theoretical advancements. First, our study extends the TPE theory by moving beyond the examination of restrictive actions and toward the exploration of corrective and promotional actions in the context of misinformation. This exploratory investigation suggests that self-other asymmetry biased perception concerning misinformation did influence individuals’ actions against misinformation, but in an unexpected direction. The results also suggest that using TPP alone to predict behavioral outcomes was deficient as it only “focuses on differences between ‘self’ and ‘other’ while ignoring situations in which the ‘self’ and ‘other’ are jointly influenced” [ 83 ]. Future research, therefore, could provide a more sophisticated understanding of third-person effects on behavior by comparing the difference of perceived effects on oneself, perceived effects on others, and the third-person perception in the pattern and strength of the effects on behavioral outcomes.

Moreover, institutionalized corrective solutions such as government and platform regulation are non-exhaustive [ 84 , 85 ]; it thus becomes critical to tap the great potential of the crowd to engage in the fight against misinformation [ 8 ] while so far, research on the motivations underlying users’ active countering of misinformation has been scarce. The current paper helps bridge this gap by exploring the role of self-efficacy and collectivism in predicting medical students’ intentions to correct misinformation and promote corrective information. We found a parallel impact of the self-ability-related factor and the collective-responsibility-related factor on intentions to correct misinformation and promote corrective information. That is, in a collectivist society like China, cultivating a sense of collective responsibility and obligation in tackling misinformation (i.e., a persuasive story told with an emphasis on collective interests of social corrections of misinformation), in parallel with systematic medical education and digital literacy training (particularly, handling various fact-checking tools, acquiring Internet skills for information seeking and verification) would be effective methods to encourage medical students to engage in active countering behaviors against misinformation. Moreover, such an effective means of encouraging social corrections of misinformation might also be applied to the general public.

In practical terms, this study lends new perspectives to the current efforts in dealing with digital misinformation by involving pre-professionals (in this case, medical students) into the fight against misinformation. As digital natives, medical students usually spend more time online, have developed sophisticated digital competencies and are equipped with basic medical knowledge, thus possessing great potential in tackling digital misinformation. This study further sheds light on how to motivate medical students to become active in thwarting digital misinformation, which can help guide strategies to enlist pre-professionals to reduce the spread and threat of misinformation. For example, collectivism education in parallel with digital literacy training would help increase medical students’ sense of responsibility for and confidence in tackling misinformation, thus encouraging them to engage in active countering behaviors.

This study also has its limitations. First, the cross-sectional survey study did not allow us to justify causal claims. Granted, the proposed direction of causality in this study is in line with extant theorizing, but there is still a possibility of reverse causal relationships. To establish causality, experimental research or longitudinal studies would be more appropriate. Our second limitation lies in the generalizability of our findings. With the focus set on medical students in Chinese society, one should be cautious in generalizing the findings to other populations and cultures. For example, the effects of collectivism on actions against misinformation might differ in Eastern and Western cultures. Further studies would benefit from replication in diverse contexts and with diverse populations to increase the overall generalizability of our findings.

Drawing on TPE theory, our study revealed that TPP failed to motivate medical students to correct misinformation and promote corrective information. However, self-efficacy and collectivism were found to serve as direct and powerful drivers of corrective and promotive actions. Accordingly, in a collectivist society such as China’s, cultivating a sense of collective responsibility in tackling misinformation, in parallel with efficient personal efficacy interventions, would be effective methods to encourage medical students, even the general public, to actively engage in countering behaviors against misinformation.

Availability of data and materials

The datasets used and/or analyzed during the current study available from the corresponding author on reasonable request.

Tencent Jiaozhen Fact-Checking Platform which comprises the Tencent information verification tool allow users to check information authenticity through keyword searching. The tool is updated on a daily basis and adopts a human-machine collaboration approach to discovering, verifying, and refuting rumors and false information. For refuting rumors, Tencent Jiaozhen publishes verified content on the homepage of Tencent's rumor-refuting platform, and uses algorithms to accurately push this content to users exposed to the relevant rumors through the WeChat dispelling assistant.

Piyao.org.cn is hosted by the Internet Illegal Information Reporting Center under the Office of the Central Cyberspace Affairs Commission and operated by Xinhuanet.com. The platform is a website that collects statements from Twitter-like services, news portals and China's biggest search engine, Baidu, to refute online rumors and expose the scams of phishing websites. It has integrated over 40 local rumor-refuting platforms and uses artificial intelligence to identify rumors.

Dhawan D, Bekalu M, Pinnamaneni R, McCloud R, Viswanath K. COVID-19 news and misinformation: do they matter for public health prevention? J Health Commun. 2021;26:799–808.

Article   PubMed   Google Scholar  

Janmohamed K, Walter N, Nyhan K, Khoshnood K, Tucker JD, Sangngam N, et al. Interventions to mitigate COVID-19 misinformation: a systematic review and meta-analysis. J Health Commun. 2021;26:846–57.

Cheng Y, Chen ZF. The influence of presumed fake news influence: examining public support for corporate corrective response, media literacy interventions, and governmental regulation. Mass Commun Soc. 2020;23:705–29.

Article   Google Scholar  

Earnshaw VA, Katz IT. Educate, amplify, and focus to address COVID-19 misinformation. JAMA Health Forum. 2020;1:e200460.

Bautista JR, Zhang Y, Gwizdka J. Healthcare professionals’ acts of correcting health misinformation on social media. Int J Med Inf. 2021;148:104375.

O’Doherty D, Lougheed J, Hannigan A, Last J, Dromey M, O’Tuathaigh C, et al. Internet skills of medical faculty and students: is there a difference? BMC Med Educ. 2019;19:39.

Article   PubMed   PubMed Central   Google Scholar  

Davison WP. The third-person effect in communication.

Koo AZ-X, Su M-H, Lee S, Ahn S-Y, Rojas H. What motivates people to correct misinformation? Examining the effects of third-person perceptions and perceived norms. J Broadcast Electron Media. 2021;65:111–34.

Oktavianus J, Bautista JR. Motivating healthcare professionals to correct online health misinformation: the roles of subjective norm, third-person perception, and channel differences. Comput Hum Behav. 2023;147:107839.

Tang S, Willnat L, Zhang H. Fake news, information overload, and the third-person effect in China. Glob Media China. 2021;6:492–507.

Chapin J. Third-person perception and facebook. Int J Cyber Behav Psychol Learn. 2014;4:34–44.

Wei R, Lo V-H, Lu H-Y. Reconsidering the relationship between the third-person perception and optimistic bias. Commun Res. 2007;34:665–84.

Weinstein ND. Unrealistic optimism about future life events. J Pers Soc Psychol. 1980;39:802–20.

Liu X. Media exposure and third-person perception: the mediating role of social realism and proxy efficacy. 2021.

Yang J, Tian Y. “Others are more vulnerable to fake news than I Am”: Third-person effect of COVID-19 fake news on social media users. Comput Hum Behav. 2021;125:106950.

Sun Ye, Shen L, Pan Z. On the behavioral component of the third-person effect. Commun Res. 2008;35:257–78.

Article   CAS   Google Scholar  

Wei R, Lo V-H. The third-person effects of political attack ads in the 2004 U.S. Presidential election. Media Psychol. 2007;9:367–88.

Duck JM, Hogg MA, Terry DJ. Social identity and perceptions of media persuasion: are we always less influenced than others? 1. J Appl Soc Psychol. 1999;29(9):1879–99.

Eveland WP, Nathanson AI, Detenber BH, McLEOD DM. Rethinking the social distance corollary: perceived likelihood of expsoure and the third-person perception. Commun Res. 1999;26:275–302.

Scharrer E. Third-person perception and television violence: the role of out-group stereotyping in perceptions of susceptibility to effects. Commun Res. 2002;29:681–704.

Brownlee K, Halverson G, Chassie A. Multiple relationships: maintaining professional identity in rural social work practice. J Compar Soc Work. 2012;7(1):81–91.

Hogg MA, Reid SA. Social identity, self-categorization, and the communication of group norms. Commun Theory. 2006;16:7–30.

Lee B, Tamborini R. Third-person effect and internet pornography: the influence of collectivism and internet self-efficacy. J Commun. 2005;55:292–310.

Wei R, Golan G. Political advertising on social media in the 2012 presidential election: exploring the perceptual and behavioral components of the third-person effect. Electron News. 2013;7:223–42.

Dong E, Du H, Gardner L. An interactive web-based dashboard to track COVID-19 in real time. Lancet Infect Dis. 2020;20:533–4. 

Jun Z, Weili W, Xin Z, Wei Z. Recommended psychological crisis intervention response to the 2019 novel coronavirus pneumonia outbreak in China: a model of West China Hospital. Precis Clin Med. 2020;3(1):3–8.

Shi Y, Zhang S, Fan L, Sun T. What motivates medical students to engage in volunteer behavior during the COVID-19 Outbreak? A large cross-sectional survey. Front Psychol. 2021;11:569765.

Passemard S, Faye A, Dubertret C, Peyre H, Vorms C, Boimare V, ... & Ricard JD. Covid-19 crisis impact on the next generation of physicians: a survey of 800 medical students. BMC Med Educ. 2021;21(1):1–13.

Tempski P, Arantes-Costa FM, Kobayasi R, Siqueira MA, Torsani MB, Amaro BQ, Martins MA. Medical students’ perceptions and motivations during the COVID-19 pandemic. PLoS ONE. 2021;16(3):e0248627.

Article   CAS   PubMed   PubMed Central   Google Scholar  

Perloff RM. Third-person effect research 1983–1992: a review and synthesis. Int J Public Opin Res. 1993;5:167–84.

Chen L, Fu L. Let’s fight the infodemic: the third-person effect process of misinformation during public health emergencies. Internet Res. 2022;32:1357–77.

Lee T. How people perceive influence of fake news and why it matters. Commun Q. 2021;69:431–53.

Liu PL, Huang LV. Digital disinformation about COVID-19 and the third-person effect: examining the channel differences and negative emotional outcomes. Cyberpsychol Behav Soc Netw. 2020;23:789–93.

Gunther AC, Thorson E. Perceived persuasive effects of product commercials and public service announcements: third-person effects in new domains. Commun Res. 1992;19:574–96.

Gunther A. What we think others think: cause and consequence in the third-person effect. Commun Res. 1991;18:355–72.

Reid SA, Hogg MA. A self-categorization explanation for the third-person effect. Hum Commun Res. 2005;31:129–61.

Cao B, Chen Z, Huang Y, Lo WH. Conflict between Mainland Chinese and Hong Kongers: a social identity perspective in explaining the hostile media phenomenon and the third-person effect. J Appl J Media Stud. 2014;3:225–40.

Google Scholar  

Tajfel H. Experimental studies of intergroup behaviour. In: Cognitive analysis of social behavior: Proceedings of the NATO advanced study Institute on “The cognitive analysis of socio-psychological processes”, Aix-enProvence, France, July 12–31, 1981 Dordrecht: Springer Netherlands; 1982. p. 227–46.

Chapter   Google Scholar  

Tajfel H, Turner JC, Austin WG, Worchel S. An integrative theory of intergroup conflict. Organizational Identity. 1979;56(65):9780203505984–9780203505916.

Crocker J, Luhtanen R. Collective self-esteem and ingroup bias. J Pers Soc Psychol. 1990;58(1).

Singelis TM. The measurement of independent and interdependent self-construals. Pers Soc Psychol Bull. 1994;20(5):580–91.

Cho H, Lee JS. The influence of self-efficacy, subjective norms, and risk perception on behavioral intentions related to the H1N1 flu pandemic: a comparison between K orea and the US. Asian J Soc Psychol. 2015;18(4):311–24.

Bandura A, Freeman WH, Lightsey R. Self-efficacy: the exercise of control. J Cogn Psychother. 1999;13:158–66.

Bandura A, Guide for constructing self-efficacy scales. Self-efficacy beliefs of adolescents. 2006;5(1):307–37.

Pajares F. Self-efficacy beliefs in academic settings. Rev Educ Res. 1996;66:543–78.

Park JS, Ahn HY, Haley EJ. Optimistic bias, advertising skepticism, and consumer intentions for seeking information about the health risks of prescription medicine. Health Mark Q. 2017;34(2):81–96.

Hofstede GH. Culture’s consequences: comparing values, behaviors, institutions, and organizations across nations. 2nd ed. Thousand Oaks: Sage Publications; 2001.

Triandis HC. Individualism and Collectivism. 1st ed. New York: Routledge; 2018.

Wated G, Sanchez JI. Managerial tolerance of nepotism: the effects of individualism-collectivism in a Latin American Context. J Bus Ethics. 2015;130:45–57.

Markus HR, Kitayama S. Culture and the self."Implications for cognition, emotion, and motivation.

Sullivan D, Landau MJ, Kay AC, Rothschild ZK. Collectivism and the meaning of suffering. J Pers Soc Psychol. 2012;103:1023–39.

Lo V, Wei R. Third-person effect, gender, and pornography on the lnternet. J Broadcast Electron Media. 2002;46:13–33.

Barnidge M, Rojas H. Hostile media perceptions, presumed media influence, and political talk: expanding the corrective action hypothesis. Int J Public Opin Res. 2014;26:135–56.

Wintterlin F, Frischlich L, Boberg S, Schatto-Eckrodt T, Reer F, Quandt T. Corrective Actions in the information disorder. the role of presumed media influence and hostile media perceptions for the countering of distorted user-generated content. Polit Commun. 2021;38:773–91.

Wei R, Lo V-H, Lu H-Y, Hou H-Y. Examining multiple behavioral effects of third-person perception: evidence from the news about Fukushima nuclear crisis in Taiwan. Chin J Commun. 2015;8:95–111.

McLEOD DM, Eveland WP, Nathanson AI. Support for censorship of violent and misogynic rap lyrics: an analysis of the third-person effect. Commun Res. 1997;24:153–74.

Nathanson AI, Eveland WP Jr, Park H-S, Paul B. Perceived media influence and efficacy as predictors of caregivers’ protective behaviors. J Broadcast Electron Media. 2002;46:385–410.

McLeod DM, Detenber BH, Eveland WP. Behind the third-person effect: differentiating perceptual processes for self and other. J Commun. 2001;51:678–95.

Rojas H. “Corrective” Actions in the public sphere: how perceptions of media and media effects shape political behaviors. Int J Public Opin Res. 2010;22:343–63.

Chung M, Munno GJ, Moritz B. Triggering participation: exploring the effects of third-person and hostile media perceptions on online participation. Comput Hum Behav. 2015;53:452–61.

Zhao L, Yin J, Song Y. An exploration of rumor combating behavior on social media in the context of social crises. Comput Hum Behav. 2016;58:25–36.

Sherman DK, Updegraff JA, Handy MS, Eom K, Kim HS. Beliefs and social norms as precursors of environmental support: the joint influence of collectivism and socioeconomic status. Pers Soc Psychol Bull. 2022;48:463–77.

Zhu Y, Wei R, Lo V-H, Zhang M, Li Z. Collectivism and altruistic behavior: a third-person effect study of COVID-19 news among Wuhan residents. Glob Media China. 2021;6:476–91.

Yang F, Horning M. Reluctant to share: how third person perceptions of fake news discourage news readers from sharing “real news” on social media. Soc Media Soc. 2020;6:205630512095517.

Adams K, Hean S, Sturgis P, Clark JM. Investigating the factors influencing professional identity of first-year health and social care students. Learn Health Soc Care. 2006;5(2):55–68.

丁汉青, 王军. 冲突与协调: 传媒从业者后备军职业认同状况研究——以北京某高校新闻学院在校生为例. 国际新闻界, 2019;2 :113–131.

Yoo B, Donthu N, Lenartowicz T. Measuring Hofstede’s five dimensions of cultural values at the individual level: development and validation of. J Int Consum Mark. 2011;23(3-4):193-210.

Tandoc EC, Lim D, Ling R. Diffusion of disinformation: how social media users respond to fake news and why. Journalism. 2020;21:381–98.

Tan ASL, Lee C, Chae J. Exposure to health (Mis)Information: lagged effects on young adults’ health behaviors and potential pathways. J Commun. 2015.

Tully M, Bode L, Vraga EK. Mobilizing users: does exposure to misinformation and its correction affect users’ responses to a health misinformation post? Soc Media Soc. 2020;6:205630512097837.

Arbuckle JL. Full information estimation in the presence of in complete data. In: Marcoulides GA, Schumaker RE, editors. Advanced structural equation modeling: issues and techniques. Mahwah: Erlbaum; 1996. p. 243–77.

Narayanan A. A review of eight software packages for structural equation modeling. Am Stat. 2012;66(2):129–38.

Sakaria D, Maat SM, Mohd Matore MEE. Examining the optimal choice of SEM statistical software packages for sustainable mathematics education: a systematic review. Sustainability. 2023;15(4):3209.

Fornell C, Larcker DF. Evaluating structural equation models with unobservable variables and measurement error. J Mark Res. 1981;18(1):39–50.

Wheaton B, Muthen B, Alwin DF, Summers GF. Assessing reliability and stability in panel models. Sociol Methodol. 1977;8:84–136.

Hu LT, Bentler PM. Cutoff criteria for fit indexes in covariance structure analysis: conventional criteria versus new alternatives. Struct Equ Modeling. 1999;6(1):1–55.

Ho SS, Goh TJ, Leung YW. Let’s nab fake science news: predicting scientists’ support for interventions using the influence of presumed media influence model. Journalism. 2022;23:910–28.

Bhagavathula AS, Aldhaleei WA, Rahmani J, Mahabadi MA, Bandari DK. Knowledge and perceptions of COVID-19 among health care workers: cross-sectional study. JMIR Public Health Surveill. 2020;6(2):e19160.

Jung EH, Zhang L, Nekmat E. SNS usage and third-person effects in the risk perception of Zika virus among Singaporean Women. J Health Commun. 2020;25:736–44.

Stavrositu CD, Kim J. Social media metrics: third-person perceptions of health information. Comput Hum Behav. 2014;35:61–7.

Wei R, Lo VH, Lu HY. Third-person effects of health news: exploring the relationships among media exposure, presumed media influence, and behavioral intentions. Am Behav Sci. 2008;52:261–77.

Hong SC. Presumed effects of “fake news” on the global warming discussion in a cross-cultural context. Sustainability. 2020;12(5).

Neuwirth K, Frederick E. Extending the framework of third-, first-, and second-person effects. Mass Commun Soc. 2002;5:113–40.

Bastick Z. Would you notice if fake news changed your behavior? An experiment on the unconscious effects of disinformation. Comput Hum Behav. 2021;116.

Harff D, Bollen C, Schmuck D. Responses to social media influencers’ misinformation about COVID-19: a pre-registered multiple-exposure experiment. Media Psychol. 2022;25:831–50.

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Acknowledgements

We thank all participants and staff working for the project.

This work was supported by Humanities and Social Sciences Youth Foundation of the Ministry of Education of China (Grant No. 21YJC860012).

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Zongya Li wrote the main manuscript and Jun yan collected the data. All authors reviewed the manuscript.

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Correspondence to Jun Yan .

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Li, Z., Yan, J. Does a perceptual gap lead to actions against digital misinformation? A third-person effect study among medical students. BMC Public Health 24 , 1291 (2024). https://doi.org/10.1186/s12889-024-18763-9

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Received : 08 December 2023

Accepted : 02 May 2024

Published : 11 May 2024

DOI : https://doi.org/10.1186/s12889-024-18763-9

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• Digital misinformation
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