This guide has everything you need on factorising and expanding the difference of two squares. Some of this blog is only relevant to the higher paper – if you’re doing the foundation paper, you don’t need to cover these parts.
We’ll start with an expansion – this will help you understand why the difference of two squares is different to other factorising questions.
Expand and simplify (𝑥 + 15)(𝑥 – 15)
Let’s use the grid method to expand this. Remember, we write one bracket across the top and the second bracket down the side. Then, we multiply each row by each column.
So (𝑥 + 15)(𝑥 – 15) = 𝑥2 + 15𝑥 – 15𝑥 – 225
= 𝑥2 – 225
If you need a reminder on this, take a look at the revision sheet on expanding double brackets. It’s in the pack below, along with exam-style questions on this topic!
You may have noticed that 225 is equal to 152. This is why the expressions are called the difference of two squares. We have one square number (225) subtracted from another squared term (𝑥2). We can use this to quickly factorise expressions in this form.
Factorise 𝑥² – 81.
The square root of 81 is 9, so 𝑥2 – 81 = (𝑥 + 9)(𝑥 – 9).
We can also compare this to the form 𝑥2 + 𝑏𝑥 + 𝑐. We have 𝑏 = 0 and 𝑐 = -81 so we are looking for a pair of numbers that multiply together to give -81 and add together to give 0, which is -9 and 9.
Higher Level Content
From here on, the content is only on the higher paper so you can stop here if you are sitting foundation.
Factorise 4𝑥² – 81.
In this case, the 𝑥2 term has a coefficient: 4. The method is just the same, except we find the square root of 4𝑥2 as well as 81. The square roots are 2𝑥 and 9 so 4𝑥2 – 81 = (2𝑥 + 9)(2𝑥 – 9).
For exam-style questions, have a look at the revision bundle below – it covers everything you need to know on expanding and factorising for your GCSE.
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