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.

Related factorising lessons

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.

• Factorising
• Factorising single brackets
• Factorising quadratics

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.

a² – b² = (a + b)(a – b)

How to factorise using 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.)

Difference of two squares examples

Example 1: x² variable has a coefficient of 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)(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

Example 3: x² variable has a coefficient greater than 1

Fully factorise

25x² – 16

Example 4: coefficient of both terms is greater than 1

Fully factorise

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)

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’

The next lessons are

• Simultaneous equations
• Solving equations
• Quadratic equations

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