Difference of two squares

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In order to access this I need to be confident with:

Expanding brackets Expand and simplify

Factors, multiples, powers and roots

Algebraic expressions

Adding and subtracting negative numbers

Multiplying and dividing negative numbers

Highest common factor (HCF)

Law of indices Square numbers and square roots

This topic is relevant for:

GCSE Maths Algebra Factorising

Difference of Two Squares

Here we will learn how to factorise using the difference of two squares method for a quadratic in the form a² – b².

There are also difference of two squares questions and an extensive worksheet based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

If you’re looking for a summary of all the different ways you can factorise expressions then you may find it helpful to start with our main factorising lesson or look in detail at the other lessons in this section.

What is the difference of two squares?

The difference of two squares is a method of factorising used when an algebraic expression includes two squared terms, one subtracted from the other:

a² – b²

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

An example of an expression we can factorise using the difference of two squares might be x² – 4 or 4x² – 25

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

What is the difference of two squares?

Difference of two squares worksheet

Get your free difference of two squares worksheet with 20+ reasoning and applied questions, answers and mark scheme.

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Difference of two squares worksheet

Get your free difference of two squares worksheet with 20+ reasoning and applied questions, answers and mark scheme.

DOWNLOAD FREE

How to factorise using difference of two squares

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

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

Explain how to factorise using the difference of two squares in 4 steps

Difference of two squares examples

Example 1: x² variable has a coefficient of 1

Fully factorise

x² – 9

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)(x – 3)

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

(x + 3)(x – 3) = x² – 9

Example 2: y² variable has a coefficient of 1

Fully factorise

64 – y²

Write down two brackets

( )( )

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

64 – y²

√64

(8 )(8 )

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

64 – y²

√(y²)

(8 y)(8 y)

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

(8 + y)(8 – y)

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

(8 – y)(8 + y) = 64 – y²

Example 3: x² variable has a coefficient greater than 1

Fully factorise

25x² – 16

Write down two brackets

( )( )

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

25x² – 16

√(25x²)

(5x )(5x )

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

25x² – 16

√16

(5x 4)(5x 4)

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

(5x + 4)(5x – 4)

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

(5x + 4)(5x – 4) = 25x² – 16

Example 4: coefficient of both terms is greater than 1

Fully factorise

4x² – 81y²

Write down two brackets

( )( )

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

4x² – 81y²

√(4x²)

(2x )(2x )

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

4x² – 81y²

√(81y²)

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

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

(2x + 9y)(2x – 9y)

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

(2x + 9y)(2x – 9y) = 4x² – 81y²

Example 5: combining HCF / GCF and difference of two squares

Fully factorise

x³ – 64x

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

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

x(x² – 64)

Write down two brackets with the x at the front

x( )( )

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

x(x² – 64)

√(x²)

x(x )(x )

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

x(x² – 64)

√64

x(x 8)(x 8)

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

x(x + 8)(x – 8)

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

x(x + 8)(x – 8) = x³ – 64x

Common misconceptions

The order of subtraction

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

( + )( – ) or ( – )( + )

Square root the entire term

Remember to square root the entire term including the coefficients of the variables.

E.g.

9x² – 49

√(9x²) = 3x

√49 = 7

Terms used can vary

The term factorising can sometimes be written as ‘factoring’ or factorization’

Practice difference of two squares questions

(x-5)(x-5)

(x+5)(x-5)

(x+1)(x-25)

(x-1)(x+25)

Recognising the expression as a difference of two squares means we can square root each term

\begin{aligned} \sqrt{x^{2}}&=x\\ \sqrt{25}&=5\\ \end{aligned}

and then rewrite as a product of two brackets

(x+5)(x-5)

(y+9)(y-9)

(y-9)(y-9)

(y+1)(y-81)

(y-1)(y+81)

Recognising the expression as a difference of two squares means we can square root each term

\begin{aligned} \sqrt{y^{2}}&=y\\ \sqrt{81}&=9\\ \end{aligned}

and then rewrite as a product of two brackets

(y+9)(y-9)

(7-y)(7-y)

(49+y)(1-y)

(7+y)(7-y)

(1+y)(49-y)

Recognising the expression as a difference of two squares means we can square root each term

\begin{aligned} \sqrt{49}&=7\\ \sqrt{y^{2}}&=y\\ \end{aligned}

and then rewrite as a product of two brackets

(7+y)(7-y)

(4+x)(1-x)

(x+2)(x-2)

(1+x)(4-x)

(2+x)(2-x)

Recognising the expression as a difference of two squares means we can square root each term

\begin{aligned} \sqrt{4}&=2\\ \sqrt{x^{2}}&=x\\ \end{aligned}

and then rewrite as a product of two brackets

(2+x)(2-x)

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

(4x-10)(4x-10)

(16x+100)(x-1)

(16x+1)(x-100)

Recognising the expression as a difference of two squares means we can square root each term

\begin{aligned} \sqrt{16x^{2}}&=4x\\ \sqrt{100}&=10\\ \end{aligned}

and then rewrite as a product of two brackets

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

(7-3y)(7-3y)

(49+y)(1-9y)

(7+9y)(7-9y)

(7+3y)(7-3y)

Recognising the expression as a difference of two squares means we can square root each term

\begin{aligned} \sqrt{49}&=7\\ \sqrt{9y^{2}}&=3y\\ \end{aligned}

and then rewrite as a product of two brackets

(7+3y)(7-3y)

(81x+y)(x-4y)

(x+4y)(81x-y)

(9x+4y)(9x-4y)

(9x-4y)(9x-4y)

Recognising the expression as a difference of two squares means we can square root each term

\begin{aligned} \sqrt{81x^{2}}&=9x\\ \sqrt{16y^{2}}&=4y\\ \end{aligned}

and then rewrite as a product of two brackets

(9x+4y)(9x-4y)

4x(x+6)(x-6)

2x(x+9)(x-9)

4x(x+3)(x-3)

4x(x-3)(x-3)

The first thing we need to do is divide by the highest common factor of the two terms and rewrite the expression

4x^{3}-36x=4x(x^{2}-9)

Recognising the expression in the bracket as a difference of two squares means we can square root each term

\begin{aligned} \sqrt{x^{2}}&=x\\ \sqrt{9}&=3\\ \end{aligned}

and then rewrite as a product of two brackets. Don’t forget to also multiply by the factor we initially divided by

4x(x+3)(x-3)

Difference of two squares GCSE questions

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)

Learning checklist

You have now learnt how to:

Manipulate algebraic expressions by taking out common factors to factorise into a single bracket
Factorise quadratic expressions of the form $x^{2}+ bx + c$
Factorise quadratic expressions of the form of the difference of two squares
Factorising quadratic expressions of the form $ax^{2} + bx + c$ (H)

The next lessons are

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