Difference of Two Squares
factorising quadratics in the form a2 – b2

We have previously learnt about factorising expressions into a single bracket and factorising quadratics into double brackets.

Here we will learn how to factorise a quadratic in the form a2 – b2, also known as the difference of two squares.

(Remember: a2 – b2 is known as a binomial because it is an expression with two terms)

There are also factorising worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is the difference of two squares?

When we have a squared term subtracted from another squared term we can use the difference of two squares method. Square numbers are sometimes called perfect squares.

To use quadratic factorisation on an expression in the form a2 – b2, we need double brackets.

Difference of two squares

a2 – b2 = (a + b)(a – b)

Difference of two squares

How to factorise quadratics: difference of two squares

In order to factorise an algebraic expression using the difference of two squares:

  1. Write down two brackets
  2. Square root the first term and write it on the left hand side of both brackets
  3. Square root the last term and write it on the right hand side of both brackets
  4. Put a + in the middle of one bracket and a – in the middle of the other (the order doesn’t matter.)

Factorising examples: difference of two squares

Example 1

Fully factorise

x2 – 9

  1. Write down two brackets

( )( )

2Square root the first term and write it on the left hand side of both brackets

x2 – 9

√x2

(x )(x )

3Square root the last term and write it on the right hand side of both brackets.

(x2 – 9)

√9

(x 3)(x 3)

4Put a + in the middle of one bracket and a – in the middle of the other (the order doesn’t matter).

(x + 3)(x3)

We can check the final answer by multiplying out the brackets!

(x + 3)(x3) = x2 – 9

Example 2

Fully factorise

64 – y2

( )( )

64 – y2

√64

(8 )(8 )

64 – y2

√y2

(8 y)(8 y)

(8 + y)(8y)

We can check the final answer by multiplying out the brackets!

(8y)(8 + y) = 64 – y2

Example 3

Fully factorise

25x2 – 16

( )( )

25x2 – 16

√25x2

(5x )(5x )

25x2 – 16

√16

(5x 4)(5x 4)

(5x + 4)(5x4)

We can check the final answer by multiplying out the brackets!

(5x + 4)(5x4) = 25x2 – 16

Example 4

Fully factorise

4x2 – 81y2

( )( )

4x2 – 81y2

√4x2

(2x )(2x )

4x2 – 81y2

√81y2

(2x 9y)(2x 9y)

(2x + 9y)(2x9y)

We can check the final answer by multiplying out the brackets!

(2x + 9y)(2x9y) = 4x2 – 81y2

Example 5

Fully factorise

x3 – 64x

Be careful, this one is not the difference of two squares!

We first need to find the greatest common factor (x) and write it outside of a single bracket.

x(x2 – 64)

x( )( )

x(x2 – 64)

√x2

x(x )(x )

x(x2 – 64)

√64

x(x 8)(x 8)

x(x + 8)(x8)

We can check the final answer by multiplying out the brackets!

x(x + 8)(x8) = x3 – 64x

Common Misconceptions

  • There must be a + in one bracket, and a – in the other, the order doesn’t matter.

    ( + )( – ) or ( – )( + )
  • Remember to square root the entire term including the coefficients of the variables.

    E.g.

    9x2 – 49

    √9x2 = 3x

    √49 = 7
  • The term factorising can sometimes be written as ‘factoring’ or factorization’

Practice Factorising Questions: difference of two squares

1. Fully factorise x2 – 25

Show answer

= (x + 5)(x – 5)

2. Fully factorise: y2 – 81

Show answer

= (y + 9)(y – 9)

3. Fully factorise: 49 – y2

Show answer

= (7 + y)(7 – y)

4. Fully factorise: 4 – x2

Show answer

= (2 + x)(2 – x)

5. Fully factorise: 16x2 – 100

Show answer

= (4x + 10)(4x -10)

6. Fully factorise: 49 – 9y2

Show answer

= (7 + 3y)(7 – 3y)

7. Fully factorise: 81x2 – 16y2

Show answer

= (9x + 4y)(9x – 4y)

8. Fully factorise: 4x3 – 36x

Show answer

= 4x(x + 3)(x – 3)

GCSE Factorising Questions: difference of two squares

1. Factorise x2 – 100

Show answer

= (x + 10)(x – 10)

(2 marks)

2. Factorise y2 – 49

Show answer

= (y + 7)(y – 7)

(2 marks)

3. Factorise 2x2 – 50

Show answer

= 2(x2 – 25) = 2(x + 5)(x – 5)

(2 marks)

Factorising worksheeets

Download for free one or more factorising worksheets to help you practise this further and more.

Learning Checklist:

You have now learnt to:

  • Manipulate algebraic expressions by taking out common factors to factorise into a single bracket.
  • Factorise quadratic expressions of the form x2 + bx + c
  • Factorise quadratic expressions of the form of the difference of two squares.
  • Factorising quadratic expressions of the form ax2 + bx + c (H)

Head back to the main Factorising page for a recap on all types of factorising, along with, mixed practice questions, mixed worksheets and exam questions.

The next topics are:

  • Solving Equations
  • Solving Linear Equations
  • Solving Quadratic Equations 
  • Quadratic Formula
  • Completing the square
  • Forming and solving quadratic equations

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