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

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  ax2+bx+c=0, where a, b, and c are real numbers and a ≠ 0.

For example:

• x2+7x+10=0

• 5x2−3x+2=0

•  −2x2+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:  ax2+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 x2+7x+12=0
Solution: Given  x2+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
∴  x2+7x+12=x2+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  x2+7x+12=0

Example 2:  3x2+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
 3x2+11x+6=32+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  3x2+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:

• a2+2ab+b2=(a+b)2

• a2−2ab+b2=(a−b)2

Example 1: Solve x2+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  x2+10x+25=0

Example 2: Solve  9x2−12x+4=0
Solution:  9x2−12x+4=  (3x)2−(3)(2)x+22=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  9x2−12x+4=0.

Factorize the Quadratic Equation using the Difference of Squares

This method is based on the identity a2−b2=(a−b)(a+b), which helps in factorising expressions quickly when they follow this pattern.

Example 1: Solve  x2−16
Solution: Given x2−16=0 is of the form a2−b2=0.
 x2−16=(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−x2=0
Solution: Given  25−x2is of the form  a2−b2=0.
 25−x2= (5 -x)(5 + x) = 0
⇒ (5 - x)(5 + x) = 0
x = 5, -5
∴ x = 5, -5 are the roots of the equation  25−x2=0.

Practise Questions on Factorization of a Quadratic Equation

1. Solve  2x2−x−6=0

2. Solve  x2+9x+20=0

3. Solve  3x2−10x+8=0

4. Solve x2−x−12=0

5. Solve 9x2–16=0

6. Solve  x2–10x+25=0