IGCSEMathsCambridge (CIE)ExtendedRevision NotesAlgebra & SequencesExpanding & Factorising BracketsDifference of Two Squares

Difference of Two Squares (Cambridge (CIE) IGCSE Maths)

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Jamie Wood

Written by: Jamie Wood

Reviewed by: Dan Finlay

Cambridge (CIE) Maths: ExtendedEdexcel Maths A (Modular): Higher Unit 1Cambridge (CIE) International Maths: Extended

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Difference of Two Squares

What is the difference of two squares?

  • When a "squared" quantity is subtracted from another "squared" quantity, you get the difference of two squares

    • For example:

      • a2 - b2

      • 92 - 52

      • (x + 1)2 - (x - 4)2

      • 4m2 - 25n2, which is (2m)2 - (5n)2

How do I factorise the difference of two squares?

  • a2 - b2 factorises to (a + b)(a - b)

    • This can be shown by expanding the brackets

      • open parentheses a plus b close parentheses open parentheses a minus b close parentheses equals a squared minus a b plus b a minus b squared equals a squared minus b squared

    • The brackets can swap order

      • a2 - b2 = (a + b)(a - b) = (a - b)(a + b)

      • (but terms inside a bracket cannot swap order)

  • For example, x squared minus 9 equals open parentheses x plus 3 close parentheses open parentheses x minus 3 close parentheses

    • This is the same as open parentheses x minus 3 close parentheses open parentheses x plus 3 close parentheses

    • But not the same as open parentheses 3 plus x close parentheses open parentheses 3 minus x close parentheses

      • which expands to 9 minus x squared

How can the difference of two squares be made harder?

  • You may find it used with:

    • numbers

      • 72 - 32 = (7+3) (7-3) = (10) (4) = 40

    • A combination of square numbers and squared variables

      • 4m2 - 9n2 = (2m)2 - (3n)2 = (2m + 3n)(2m - 3n)

    • Any other powers which can be written as a difference of two squares

      • a4 - b4 = (a2)2 - (b2)2 = (a2 + b2) (a2 - b2)

      • r8 - t6 = (r4)2 - (t3)2 = (r4 + t3) (r4 - t3)

  • You may also need to take out a common factor first

    • 2 x squared minus 18 equals 2 open parentheses x squared minus 9 close parentheses giving 2 open parentheses x plus 3 close parentheses open parentheses x minus 3 close parentheses

      • The 2 comes out in front

Can I use the difference of two squares to expand?

  • Using the difference of two squares to expand is quicker than expanding double brackets and collecting like terms

  • Brackets of the form (a + b)(a - b) expand to a2 - b2

    • For example open parentheses 2 x plus 3 close parentheses open parentheses 2 x minus 3 close parentheses expands to 4 x squared minus 9

Examiner Tips and Tricks

  • The difference between two squares is often the trick required to complete a harder algebraic question in the exam

    • Make sure you are able to spot it!

Worked Example

(a) Factorise  9 x squared minus 16.

Recognise that 9 x squared and 16 are both squared terms

Therefore you can factorise using the difference of two squares

Rewrite as a difference of two squared terms

9 x squared minus 16 equals open parentheses 3 x close parentheses squared minus open parentheses 4 close parentheses squared

Use the rule a squared minus b squared equals open parentheses a plus b close parentheses open parentheses a minus b close parentheses

stretchy left parenthesis 3 x plus 4 stretchy right parenthesis stretchy left parenthesis 3 x minus 4 stretchy right parenthesis 

(b) Factorise 4 r squared minus t to the power of 4.

Recognise that 4 r squared and t to the power of 4 are both squared terms

Therefore you can factorise using the difference of two squares

Rewrite as a difference of two squared terms

4 r squared minus t to the power of 4 equals open parentheses 2 r close parentheses squared minus open parentheses t squared close parentheses squared

Use the rule a squared minus b squared equals open parentheses a plus b close parentheses open parentheses a minus b close parentheses

stretchy left parenthesis 2 r plus t squared stretchy right parenthesis stretchy left parenthesis 2 r minus t squared stretchy right parenthesis

(c) Factorise 2 y squared minus 50

This does not appear to be in the form a squared minus b squared

There is a common factor of 2, so take this factor out

2 open parentheses y squared minus 25 close parentheses

You can now see y squared minus 25 which has the form y squared minus 5 squared

Use the rule a squared minus b squared equals open parentheses a plus b close parentheses open parentheses a minus b close parentheses

y squared minus 25 equals open parentheses y plus 5 close parentheses open parentheses y minus 5 close parentheses

Now multiply this answer by 2 (leaving the 2 on the outside)

2 open parentheses y plus 5 close parentheses open parentheses y minus 5 close parentheses

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    Author
    Reviewer
    Jamie Wood

    Author: Jamie Wood

    Expertise: Maths

    Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.

    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.