The Difference of Two Squares

Mentioned specifically in the Edexcel Advance Information for the 2022 GCSE exams is something called the Difference of Two Squares. So I decided to write a blog about it.

I’ve already touched on factorising quadratics in these two blogs – Factorising Quadratics (Foundation) and Factorising Quadratics (Higher). In those two blogs, the quadratics were factorised into two brackets. The general rule is that if the expression has three terms it factorises into two brackets and if it has two terms it factorises into one bracket. See the examples below:

x² + 10x = x(x +10)                            two terms factorise into one bracket
x² + 2x – 24 = (x – 4)(x + 6)                three terms factorise into two brackets
12x + 14 = 2(6x + 7)                          two terms factorise into one bracket
2x² + 7x + 3 = (2x + 1)(x + 3)             three terms factorise into two brackets

However, there is one special case where this rule does not work. This special case is called the Difference of Two Squares.

What is the Difference of Two Squares?

The Difference of Two Squares is a special type of quadratic expression where two terms actually factorise into two brackets. Below is an example.

x² – 9

This expression has two terms, but if you try to put it into a single bracket, you will not be able to – it’s impossible.

It actually factorises to this:

(x – 3)(x + 3)

Now, you can get this result by using the double-bracket method for factorising quadratics – you can write this as a three-term quadratic with no x-value, like below:

x² + 0x – 9

You are looking for two numbers that multiply together to get -9 and add together to 0.

3 × -3 = -9 and -3 + 3 = 0

How do I spot the Difference of Two Squares?

To spot a difference of two squares problem, you need to look out for a two-term quadratic, where one term is an x² term and the other is just a number and there is a minus in between them (this only works when there is a minus).

Another example:

x² – 36

This is an x² term and a number (36) and there is a minus in between.

In general terms, a quadratic of the form a² – b² factorises to (a + b)(a – b). So, basically it’s:

(square root of 1st term + square root of 2nd term)(square root of 1st term – square root of 2nd term)

So,

x² – 36 = (x + 6)(x – 6)

Some further examples

This rule generally appears in questions where both numbers can be square rooted easily, but it could also come up with non-square numbers, and the answer will include a surd. Like below:

This rule technically works for any two terms, you just need to square root both terms. So it also works for some more complex examples:

Test Your Understanding

Now let’s look at a few more examples of quadratic expressions, and determine whether they qualify for the Difference of Two Squares (and how many brackets they factorise into):

x² – 16x

You can take a factor of x out so it can go into one bracket. It factorises to x(x – 16)

x² + 11x + 24

This has three terms, so has to go into two brackets. It factorises to (x + 3)(x + 8)

x² – 9

This is a difference of two squares example, as it has an x² and a number. It factorises to (x + 3)(x – 3)

4x² – 4x + 1

This has three terms, so has to go into two brackets. It factorises to (2x – 1)(2x – 1)

16x² – 25

This is a difference of two squares example. It factorises to (4x + 5)(4x – 5)

4x⁴ + 16y²

This does not qualify for the difference of two squares because there is a plus and not a minus. You can however take a factor of 4 out. It factorises to 4(x⁴ + 4y²)

9x² – 100y²

This is a difference of two squares example. It factorises to (3x + 10y)(3x – 10y)

x² + 16

This is a bit of a red herring. It doesn’t qualify for the difference of two squares because there is a plus not a minus. This one actually cannot be factorised at all.



I hope you found this useful. This topic is specifically mentioned in the advance information for the Summer 2022 Edexcel Higher Tier exam, so we know for sure something in this blog will come up!

You can read the blog about the Summer 2022 Advance Information here.

You can practice your factorising skills in our factorising worksheet, or on our algebraic fractions worksheet. Also, CorbettMaths have written a worksheet that specifically focuses on the difference of two squares so that would be a really good resource to work through.

If you are in Bristol and need someone to show this to you in person, book a free taster session.

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