Which method is best for solving quadratic equations?

Use factorisation when the middle term splits easily. Use the quadratic formula when factorisation is difficult or when the question asks for the nature of roots.

What is the discriminant?

The discriminant is D = b^2 - 4ac. It tells whether the roots are real and distinct, real and equal, or not real.

CBSE Class 10 Maths Chapter 4 Notes: Quadratic Equations

Q: What is a quadratic equation?

A quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b, and c are real numbers and a is not equal to 0.

Exercise Solutions Available

Step-by-step answers for all textbook questions in this chapter

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Chapter Overview

Quadratic Equations is one of the most scoring Algebra chapters in Class 10 Maths. It is important because the same idea appears later in graphs, coordinate geometry, applications, and higher classes.

This chapter teaches:

What a quadratic equation is
How to solve by factorisation
How to solve by completing the square
How to use the quadratic formula
How to use discriminant to identify the nature of roots
How to convert word problems into quadratic equations

1. Standard Form of a Quadratic Equation

The standard form is:

ax^2 + bx + c = 0

where:

a, b, and c are real numbers
a != 0
x is the variable

Examples

Equation	Is it quadratic?	Reason
x^2 + 5x + 6 = 0	Yes	Highest power is 2
3x^2 - 7 = 0	Yes	Highest power is 2
4x + 9 = 0	No	Highest power is 1
x^3 + x + 1 = 0	No	Highest power is 3

2. Roots of a Quadratic Equation

A value of x that satisfies the equation is called a root or solution.

Example:

x^2 - 5x + 6 = 0

Try x = 2:

2^2 - 5(2) + 6 = 4 - 10 + 6 = 0

So, 2 is a root.

Try x = 3:

3^2 - 5(3) + 6 = 9 - 15 + 6 = 0

So, 3 is also a root.

3. Solving by Factorisation

This is the fastest method when factors are easy.

Steps

1. Write the equation in standard form. 2. Split the middle term. 3. Take common factors from pairs. 4. Factorise completely. 5. Put each factor equal to zero.

Solved Example

Solve:

x^2 + 7x + 12 = 0

Find two numbers whose product is 12 and sum is 7.

They are 3 and 4.

x^2 + 3x + 4x + 12 = 0
x(x + 3) + 4(x + 3) = 0
(x + 3)(x + 4) = 0

So:

x + 3 = 0 or x + 4 = 0
x = -3 or x = -4

Answerx = -3, -4.

4. Quadratic Formula

For:

ax^2 + bx + c = 0

the roots are:

x = (-b +/- sqrt(b^2 - 4ac)) / 2a

This method works for all quadratic equations where real roots exist.

Solved Example

Solve:

2x^2 - 5x + 3 = 0

Here:

a = 2, b = -5, c = 3

Apply formula:

x = (-(-5) +/- sqrt((-5)^2 - 4(2)(3))) / 2(2)
x = (5 +/- sqrt(25 - 24)) / 4
x = (5 +/- 1) / 4

So:

x = 6/4 = 3/2
x = 4/4 = 1

Answerx = 1, 3/2.

5. Discriminant and Nature of Roots

The expression:

D = b^2 - 4ac

is called the discriminant.

Discriminant	Nature of Roots
D > 0	Two distinct real roots
D = 0	Two equal real roots
D < 0	No real roots

Example

Find the nature of roots of:

x^2 - 4x + 4 = 0

Here:

a = 1, b = -4, c = 4
D = b^2 - 4ac
D = (-4)^2 - 4(1)(4)
D = 16 - 16
D = 0

So the roots are real and equal.

6. Completing the Square

This method converts a quadratic expression into a perfect square form.

Example

Solve:

x^2 + 6x + 5 = 0

Move constant:

x^2 + 6x = -5

Take half of coefficient of x, square it, and add to both sides.

Half of 6 is 3, and 3^2 = 9.

x^2 + 6x + 9 = -5 + 9
(x + 3)^2 = 4
x + 3 = +/- 2

So:

x = -1 or x = -5

7. Word Problems

Word problems are the most important part of this chapter. The marks are in equation formation.

Method

1. Read what has to be found. 2. Let the unknown be x. 3. Convert the condition into an equation. 4. Bring it to standard form. 5. Solve. 6. Reject impossible values if the question is about age, length, speed, or number of objects.

Example: Consecutive Integers

The product of two consecutive positive integers is 306. Find the integers.

Let the integers be:

x and x + 1

Given:

x(x + 1) = 306
x^2 + x - 306 = 0

Factorise:

x^2 + 18x - 17x - 306 = 0
x(x + 18) - 17(x + 18) = 0
(x + 18)(x - 17) = 0

So:

x = -18 or x = 17

Since positive integers are required, reject -18.

Answer17 and 18.

8. Important Question Types

Question Type	Best Method
Simple equation with easy factors	Factorisation
Equation with difficult factors	Quadratic formula
Nature of roots	Discriminant
Form equation from story	Define variable first
Equal roots condition	Put D = 0
No real roots condition	Show D < 0

9. Practice Questions

1. Solve x^2 - 8x + 15 = 0. 2. Solve 2x^2 + x - 6 = 0. 3. Find the nature of roots of 3x^2 - 2x + 1 = 0. 4. Find the value of k if x^2 + 4x + k = 0 has equal roots. 5. The product of two consecutive positive integers is 420. Find the integers. 6. The sum of squares of two consecutive odd positive integers is 394. Find them. 7. Solve 4x^2 - 4ax + (a^2 - b^2) = 0.

10. Quick Answers

1. x = 3, 5. 2. x = 3/2, -2. 3. D = -8, so no real roots. 4. D = 0; 16 - 4k = 0; k = 4. 5. 20 and 21. 6. 13 and 15. 7. x = (a + b)/2, (a - b)/2.

Common Mistakes to Avoid

Forgetting that a cannot be zero in a quadratic equation.
Splitting the middle term with wrong signs.
Writing b^2 - 4ac incorrectly when b is negative.
Forgetting +/- in the quadratic formula.
Not rejecting negative answers in real-life word problems.
Giving only roots without showing the equation formation.

Final Revision Checklist

I can identify a quadratic equation.
I can solve by factorisation.
I can apply the quadratic formula correctly.
I can find the discriminant and nature of roots.
I can form equations from word problems.
I can reject impossible values in application questions.

Internal Links

[CBSE Class 10 Maths guide](/school/cbse/class/10/maths)
[Real Numbers notes](/school/cbse/class/10/maths/cbse-real-numbers)
[Practice aptitude questions](/aptitude)

Frequently Asked Questions

A quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b, and c are real numbers and a is not equal to 0.