Difference of two squares

Tier: #Higher

🔗What you need to know first

How to

Recognising quadratic expressions in the form $ax^2-by^2$ where a and b are square numbers NB y will likely equal 1 to leave $ax^2-b$.

e.g. $x^2-36=(x+6)(x-6)$ $4x^2-25=(2x+5)(2x-5)$ $9x^{2}-16y^{2}=(3x+4y)(3x-4y)$

The difference of two squares (or DOTS as it is commonly referred) is a higher level concept which you are expected to recognise without any prompting. It crops up in several places, most notably Factorising quadratics, Algebraic fractions and Surds.

Common topics that go alongside

Factorising quadratics

❓Questions to practise

Practise these questions →

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📝Past paper questions

2024 Nov 1H GCSE Q11 (1 mark)

2024 Jun 3H GCSE Q15 (1 mark)

2023 Jun 3H GCSE Q15 (1 mark)

2022 Jun 3H GCSE Q14 (2 marks)

2021 Nov 3H GCSE Q22 (1 mark)

2019 Nov 1H GCSE Q16 (1 mark)

2019 Nov 3H GCSE Q22 (1 mark)

2018 Nov 1H GCSE Q10 (2 marks)

2018 Jun 1H GCSE Q15 (2 marks)

2018 Nov 2H GCSE Q12 (1 mark)

2017 Jun 3H GCSE Q14 (1 mark)

💬What the examiners say

"Fewer students realised the need to factorise the denominator of the first fraction and much fruitless algebra was seen with students frequently multiplying out expressions."

⬆️How you can quickly improve

After every factorisation, check whether the bracket that remains can be factorised further — specifically look for a² − b² patterns.
When dividing algebraic fractions, factorise every numerator and denominator before doing anything else, then cancel common factors.
Practise spotting difference of two squares in both algebraic and numerical contexts — the pattern is always (something)² − (something else)².

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