Factorising Simple Quadratics (Edexcel GCSE Maths) : Revision Note

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Factorising Simple Quadratics

What is a quadratic expression?

  • A quadratic expression is in the form:

    • ax2 + bx + c (where a ≠ 0)

  • If there are any higher powers of x (like x3 say) then it is not a quadratic

How do I factorise quadratics by inspection?

  • This is shown most easily through an example: factorising x squared minus 2 x minus 8

  • We need a pair of numbers that for x squared plus b x plus c

    • multiply to give c

      • which in this case is -8

    • and add to give b

      • which in this case is -2

    • +2 and -4 satisfy these conditions

      • 2 × (-4) = -8  and  2 + (-4) = -2

    • Write these numbers in a pair of brackets like this: 

      • open parentheses x plus 2 close parentheses open parentheses x minus 4 close parentheses

How do I factorise quadratics by grouping?

  • This is shown most easily through an example: factorising x squared minus 2 x minus 8

  • We need a pair of numbers that for x squared plus b x plus c

    • multiply to give c

      • which in this case is -8

    • and add to give b

      • which in this case is -2

    • +2 and -4 satisfy these conditions

      • 2 × (-4) = -8  and  2 + (-4) = -2

    • Rewrite the middle term by using +2x and -4x

      • x squared plus 2 x minus 4 x minus 8

    • Group and factorise the first two terms, using x as the common factor

    • and group and factorise the last two terms, using -4 as the common factor

      • x open parentheses x plus 2 close parentheses minus 4 open parentheses x plus 2 close parentheses

    • Note that these both now have a common factor of (x + 2) so this whole bracket can be factorised out

      • open parentheses x plus 2 close parentheses open parentheses x minus 4 close parentheses

How do I factorise quadratics using a grid?

  • This is shown most easily through an example: factorising x squared minus 2 x minus 8

  • We need a pair of numbers that for x squared plus b x plus c

    • multiply to give c

      • which in this case is -8

    • and add to give b

      • which in this case is -2

    • +2 and -4 satisfy these conditions

      • 2 × (-4) = -8  and  2 + (-4) = -2

    • Write the quadratic equation in a grid (as if you had used a grid to expand the brackets)

      • splitting the middle term as +2x and -4x

  • The grid works by multiplying the row and column headings, to give a product in the boxes in the middle

 

 

 

 

x2

-4x

 

+2x

-8

  • Write a heading for the first row, using x as the highest common factor of x2 and -4x

 

 

 

x

x2

-4x

 

+2x

-8

  • You can then use this to find the headings for the columns

    • e.g. “What does x need to be multiplied by to give x2?”

    • and “What does x need to be multiplied by to give -4x?”

 

x

-4

x

x2

-4x

 

+2x

-8

  • We can then fill in the remaining row heading using the same idea

    • e.g. “What does x need to be multiplied by to give +2x?”

    • or “What does -4 need to be multiplied by to give -8?”

 

x

-4

x

x2

-4x

+2

+2x

-8

  • We can now read off the factors from the column and row headings

    • open parentheses x plus 2 close parentheses open parentheses x minus 4 close parentheses

Which method should I use for factorising simple quadratics?

  • The first method, by inspection, is by far the quickest

    • So this is recommended in an exam for simple quadratics (where a = 1)

  • However the other two methods (grouping, or using a grid) can be used for harder quadratic equations where ≠ 1

    • So you should learn at least one of them too

Examiner Tips and Tricks

As a check, expand your answer and make sure you get the same expression as the one you were trying to factorise.

Worked Example

(a) Factorise x squared minus 4 x minus 21.

We will factorise by inspection

We need two numbers that multiply to give -21, and sum to give -4
+3 and -7 satisfy this

3 cross times open parentheses negative 7 close parentheses equals negative 21

3 plus open parentheses negative 7 close parentheses equals negative 4

Write down the brackets

 (x + 3)(x - 7)

(b) Factorise x squared minus 5 x plus 6.

We will factorise by splitting the middle term and grouping

We need two numbers that multiply to 6, and sum to -5
-3 and -2 satisfy this

open parentheses negative 3 close parentheses cross times open parentheses negative 2 close parentheses equals 6

open parentheses negative 3 close parentheses plus open parentheses negative 2 close parentheses equals negative 5

Split the middle term

x2 - 2x - 3x + 6

Factorise x out of the first two terms

x(x - 2) - 3x +6

Factorise -3 out of the last two terms

x(x - 2) - 3(x - 2)

These have a common factor of (x - 2) which can be factored out

(x - 2)(x - 3)

(c) Factorise x squared minus 2 x minus 24.

We will factorise by using a grid

We need two numbers that multiply to -24, and sum to -2
+4, and -6 satisfy this

4 cross times open parentheses negative 6 close parentheses equals negative 24

4 plus open parentheses negative 6 close parentheses equals negative 2

Use these to split the -2x term and write in a grid

 

 

 

 

x2

+4x

 

-6x

-24

Write a heading using a common factor for the first row

 

 

 

x

x2

+4x

 

-6x

-24

 Work out the headings for the rows
“What does x need to be multiplied by to make x2?”
“What does x need to be multiplied by to make +4x?”

 

x

+4

x

x2

+4x

 

-6x

-24

Repeat for the heading for the remaining row
“What does x need to be multiplied by to make -6x?”
(Or “What does +4 need to be multiplied by to make -24?”)

 

x

+4

x

x2

+4x

-6

-6x

-24

Read off the factors from the column and row headings

(x + 4)(x - 6)

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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 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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