National 5MathsSQARevision NotesAlgebraic SkillsFactorisationDifference of Two Squares

Difference of Two Squares (SQA National 5 Maths): Revision Note

Exam code: X847 75

Roger B

Written by: Roger B

Reviewed by: Dan Finlay

Updated on

Factorising using 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 be written in either 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

What are some trickier examples of difference of two squares?

  • Difference of two squares can be 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 brackets?

  • 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

    • Recognising that is quicker than expanding to 4 x squared minus 6 x plus 6 x minus 9 first and then simplifying

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.

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

Answer:

Part (a)

Recognise that 9 x squared equals open parentheses 3 x close parentheses squared and 16 equals 4 squared 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 

Part (b)

Recognise that 4 r squared equals open parentheses 2 r close parentheses squared and t to the power of 4 equals open parentheses t squared close parentheses squared 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

Worked Example

Factorise fully 2 y squared minus 50.

Answer:

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

However there is a common factor of 2, so take this factor out

2 y squared minus 50 equals 2 open parentheses y squared minus 25 close parentheses

Now you can 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

Therefore

table row cell 2 y squared minus 50 end cell equals cell 2 open parentheses y squared minus 25 close parentheses end cell row blank equals cell 2 open parentheses y plus 5 close parentheses open parentheses y minus 5 close parentheses end cell end table

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

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    Author
    Reviewer
    Roger B

    Author: Roger B

    Expertise: Maths Content Creator

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject 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.