GCSEMathsEdexcelHigherRevision Notes2. AlgebraFactorisingDeciding the Factorisation Method

Deciding the Factorisation Method (Edexcel GCSE Maths): Revision Note

Jamie Wood

Written by: Jamie Wood

Reviewed by: Dan Finlay

Updated on

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Quadratics Factorising Methods

How do I know if an expression factorises?

  • The easiest way to check if ax2 + bx + c factorises is to check if you can find a pair of integers which:

    • Multiply to give ac

    • Sum to give b

    • If you can find integers to satisfy this, the expression must factorise

  • There are some alternate methods to check:

    • Method 1: Use a calculator to solve the quadratic expression equal to 0

      • Only some calculators have this functionality

      • If the solutions are integers or fractions (without square roots), then the quadratic expression will factorise

    • Method 2: Find the value under the square root in the quadratic formula

      • b2 – 4ac

      • If this number is a square number, then the quadratic expression will factorise

Which factorisation method should I use for a quadratic expression?

  • Does it have 2 terms only?

    • Yes, like x squared minus 7 x

      • Factorise out the highest common factor, x

      • x open parentheses x minus 7 close parentheses

    • Yes, like x squared minus 9

      • Use the "difference of two squares" to factorise

      • open parentheses x plus 3 close parentheses open parentheses x minus 3 close parentheses

Does it have 3 terms?

  • Yes, starting with x2 like x squared minus 3 x minus 10

    • Use "factorising simple quadratics" by finding two numbers that add to -3 and multiply to -10

    • open parentheses x plus 2 close parentheses open parentheses x minus 5 close parentheses

  • Yes, starting with ax2 like 3 x squared plus 15 x plus 18

    • Check to see if the 3 in front of x2 is a common factor for all three terms (which it is in this case), then factorise it out of all three terms

    • 3 open parentheses x squared plus 5 x plus 6 close parentheses

    • The quadratic expression inside the brackets is now x2 +... , which factorises more easily

    • 3 open parentheses x plus 2 close parentheses open parentheses x plus 3 close parentheses

  • Yes, starting with ax2 like 3 x squared minus 5 x minus 2

    • The 3 in front of x2 is not a common factor for all three term

    • Use "factorising harder quadratics", for example factorising by grouping or factorising using a grid

    • open parentheses 3 x plus 1 close parentheses open parentheses x minus 2 close parentheses

What other expressions should I be able to factorise?

  • You may have a cubed term like x cubed minus 3 x squared minus 10 x

    • Check to see if x is a common factor for all three terms (which it is in this case), so factorise it out of all three terms

    • x open parentheses x squared minus 3 x minus 10 close parentheses

    • The remaining quadratic can then be factorised

    • x open parentheses x plus 2 close parentheses open parentheses x minus 5 close parentheses

  • It can also be useful to spot a quadratic in the form x squared plus 2 a x plus a squared

    • This factorises to open parentheses x plus a close parentheses squared

    • E.g. x squared plus 6 x plus 9 space equals space open parentheses x plus 3 close parentheses squared

Examiner Tips and Tricks

  • A common mistake in the exam is to divide expressions by numbers, e.g. 2 x squared plus 4 x plus 2 becomes x squared plus 2 x plus 1 (which is incorrect)

    • This can only be done with equations

    • e.g. 2 x squared plus 4 x plus 2 equals 0 becomes x squared plus 2 x plus 1 equals 0 (dividing "both sides" by 2)

Worked Example

Factorise  negative 8 x squared plus 100 x minus 48.

Spot the common factor of -4 and factorise it out

negative 8 x squared plus 100 x minus 48 equals negative 4 open parentheses 2 x squared minus 25 x plus 12 close parentheses

Check to see if the quadratic in the bracket will factorise using b squared minus 4 a c

table row blank blank cell open parentheses negative 25 close parentheses squared minus open parentheses 4 cross times 2 cross times 12 close parentheses end cell row blank equals cell 625 minus 96 end cell row blank equals 529 end table

529 is a square number (232) so the expression will factorise

Factorise 2 x squared minus 25 x plus 12

We require a pair of numbers which multiply to ac, and sum to b

a cross times c equals 2 cross times 12 equals 24

The only numbers which multiply to 24 and sum to -25 are

-24 and -1

Split the negative 25 x term into negative 24 x minus x

table row blank blank cell 2 x squared minus 24 x minus x plus 12 end cell end table

Group and factorise the first two terms, using 2 x as the common factor
Group and factorise the last two terms using negative 1 as the common factor

table row blank blank cell 2 x open parentheses x minus 12 close parentheses minus 1 open parentheses x minus 12 close parentheses end cell end table

These factorised terms now have a common term of open parentheses x minus 12 close parentheses, so this can be factorised out

open parentheses 2 x minus 1 close parentheses open parentheses x minus 12 close parentheses

Recall that -4 was factorised out at the start

negative 8 x squared plus 100 x minus 48 equals negative 4 open parentheses 2 x squared minus 25 x plus 12 close parentheses equals negative 4 open parentheses 2 x minus 1 close parentheses open parentheses x minus 12 close parentheses

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Author
Reviewer
Jamie Wood

Author: Jamie Wood

Expertise: Maths Content Creator

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

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.

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