Class 10 - Solution of Quadratic Equation by Factorisation

Solving a quadratic equation by factorisation involves breaking down a quadratic expression into two linear factors and then finding the values of the variable that satisfy the equation. This method is useful when the equation can be easily factorised, saving time and effort compared to other techniques that are used to find the roots.  In this guide, you’ll learn clear step-by-step methods, useful formulas, and practical examples to help you confidently solve problems involving quadratic equations.

Table of Contents

• What are Quadratic Equations
• Factorization of a Quadratic Equation
• Factorization of a Quadratic Equation using Splitting the Middle Term
• Factorization of a Quadratic Equation using Perfect Square Trinomial
• Factorization of a Quadratic Equation using the Difference of Squares
• Practise Questions on Factorization of a Quadratic Equation
• Frequently Asked Questions on Solution of Quadratic equation by Factorisation

What are Quadratic Equations

Any equation of the form p(x) = 0, where p(x) is a polynomial of degree 2 is a quadratic equation.

Standard form: A quadratic equation in the variable x is an equation of the form  $ax^{2} + bx + c = 0$, where a, b, and c are real numbers and a ≠ 0.

For example:

• $x^{2} + 7x + 10= 0$

• $5x^{2} - 3x + 2 = 0$

•  $-2x^{2} + 4x - 3 = 0$
  
  

Read more: Important Questions on Quadratic Equations - Class 10

Factorization of a Quadratic Equation

Factorisation means expressing a quadratic equation as a product of two linear factors:  $ax^{2}+bx+c=(x- 𝛼 )(x-𝛽)$

By factorising, we solve the equation by applying the zero product property: (x-𝛼)(x-𝛽) = 0

⇒(x-𝛼) = 0 or (x-𝛽) = 0

This gives the roots of the equation, x = 𝛼, 𝛽.

Read more: Quadratic Equations

Factorize the Quadratic Equation using Splitting the Middle Term

We can use the following steps to solve the quadratic equations by factoring:

1. Write the equation in standard form

2. Multiply a × c 

3. Find two numbers such that 

• Sum = b

• Product = ac

4. Split middle term

5. Factor by grouping

6. Apply the zero product property to find the roots

Example 1: Solve $x^{2}+7x+12=0$
Solution: Given  $x^{2}+7x+12=0,$ a = 1, b = 7 and c = 12.
Product = ac =  1 × 12 = 12 , sum = b = 7
Numbers 3 and 4 satisfy both conditions  3 × 4 = 12 and 3 + 4 = 7
∴  $x^{2}+7x+12 = x^{2}+3x + 4x +12 = 0$

⇒ x (x + 3) + 4(x + 3) = 0
⇒ (x + 3)(x +4) = 0
⇒ x +3 = 0 or x + 4 = 0
⇒ x = -3 or x = -4
∴  x = -3, -4 are the roots of the equation  $x^{2}+7x+12=0$

Example 2:  $3x^{2}+11x+6=0$
Solution: Given a = 3, b = 11, and c = 6,
Product = ac = 3 × 6 = 18, sum = b = 11. Numbers 9 and 2 satisfy both conditions: 9 × 2 = 18, and 9 + 2 = 11
 $3x^{2}+11x+6 = 3^{2}+9x+2x+6 = 0$
⇒  3x(x+3)+2(x+3)
⇒ (3x+2)(x+3)=0
⇒ x = -2/3 or x = -3
∴ x = −2/3, −3 are the roots of the equation  $3x^{2}+11x+6=0$

Factorize the Quadratic Equation using Perfect Square Trinomial

The method of converting any quadratic polynomial into a perfect square is known as the perfect square trinomial method.

The following equations are the perfect square trinomial formulas:

• $a^{2} + 2ab + b^{2} = (a + b)^{2}$

• $a^{2} - 2ab + b^{2} = (a - b)^{2}$

Example 1: Solve $x^{2}+10x+25 = 0$
Solution: x+10x+25 = $(x+5)^{2} = 0$
 $(x+5)^{2} = 0$⇒ (x+5) = 0 or (x+5) = 0
x = -5,-5
∴ x =-5,-5 are the roots of the equation  $x^{2}+10x+25 = 0$

Example 2: Solve  $9x^{2}-12x+4 = 0$
Solution:  $9x^{2}-12x+4$=  $(3x)^{2}-(3)(2)x + 2^{2}=0$
⇒  $(3x-2)^{2}$= 0
⇒(3x-2) = 0 or (3x-2) = 0
x = 2/3, 2/3
∴ x = 2/3, 2/3 are the roots of the equation  $9x^{2}-12x+4 = 0.$

Factorize the Quadratic Equation using the Difference of Squares

This method is based on the identity $a^{2}−b^{2}=(a−b)(a+b)$, which helps in factorising expressions quickly when they follow this pattern.

Example 1: Solve  $x^{2}−16$
Solution: Given $x^{2}−16 = 0$ is of the form $a^{2}−b^{2} = 0.$
 $x^{2}−16 = (x - 4)(x + 4) = 0$
⇒ (x - 4)(x + 4) = 0
x = 4, -4
∴ x = 4, -4 are the roots of the equation x^{2}−16 = 0.

Example 2: $25 -  x^{2} = 0$
Solution: Given  $25 -  x^{2}$is of the form  $a^{2}−b^{2} = 0$.
 $25 -  x^{2}$= (5 -x)(5 + x) = 0
⇒ (5 - x)(5 + x) = 0
x = 5, -5
∴ x = 5, -5 are the roots of the equation  $25 -  x^{2} = 0.$

Practise Questions on Factorization of a Quadratic Equation

1. Solve  $2x^{2}−x−6=0$

2. Solve  $x^{2}+9x+20=0$

3. Solve  $3x^{2}−10x+8=0$

4. Solve $x^{2}−x−12=0$

5. Solve $9x^{2} – 16 = 0$

6. Solve  $x^{2} – 10x + 25 = 0$