Factorising Harder Quadratics (Edexcel IGCSE Maths A (Modular)) : Revision Note

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Factorising Harder Quadratics

How do I factorise a quadratic expression where a ≠ 1 in ax2 + bx + c?

Method 1: Factorising by grouping

  • This is shown most easily through an example: factorising 4 x squared minus 25 x minus 21

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

    • both multiply to give ac

      • ac in this case is 4 × -21 = -84

    • and both add to give b

      • b in this case is -25

    • -28 and +3 satisfy these conditions

    • Rewrite the middle term using -28x and +3x

      • 4 x squared minus 28 x plus 3 x minus 21

    • Group and fully factorise the first two terms, using 4x as the common factor

    • and group and fully factorise the last two terms, using 3 as the common factor

      • 4 x open parentheses x minus 7 close parentheses plus 3 open parentheses x minus 7 close parentheses

    • These terms now have a common factor of open parentheses x minus 7 close parentheses

      • This whole bracket can be factorised out

      • This gives the answer open parentheses x minus 7 close parentheses open parentheses 4 x plus 3 close parentheses

Method 2: Factorising using a grid

  • Use the same example: factorising 4 x squared minus 25 x minus 21

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

    • multiply to give ac

      • ac in this case is 4 × -21 = -84

    • and add to give b

      • b in this case is -25

    • -28 and +3 satisfy these conditions

    • Write the quadratic equation in a grid

      • (as if you had used a grid to expand the brackets)

      • splitting the middle term up as -28x and +3x (either order)

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

 

 

 

 

4x2

-28x

 

+3x

-21

  • Write a heading for the first row, using 4x as the highest common factor of 4x2 and -28x

 

 

 

4x

4x2

-28x

 

+3x

-21

  • You can then use this to find the headings for the columns, e.g. “What does 4x need to be multiplied by to give 4x2?”

 

x

-7

4x

4x2

-28x

 

+3x

-21

  • 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 +3x?”

 

x

-7

4x

4x2

-28x

+3

+3x

-21

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

    • open parentheses x minus 7 close parentheses open parentheses 4 x plus 3 close parentheses

Worked Example

(a) Factorise 6 x squared minus 7 x minus 3.

We will factorise by grouping

We need two numbers that:

multiply to 6 × -3 = -18
and sum to -7

-9, and +2

Split the middle term up using these values

6x2 + 2x - 9x - 3

Factorise 2x out of the first two terms

2x(3x + 1) - 9x - 3

Factorise -3 of out the last two terms

2x(3x + 1) - 3(3x + 1)

These have a common factor of (3x + 1) which can be factorised out

(3x + 1)(2x - 3)

(b) Factorise 10 x squared plus 9 x minus 7.

We will factorise using a grid

We need two numbers that:

multiply to 10 × -7 = -70
and sum to +9

-5, and +14

Use these values to split the 9x term and write in a grid

10x2

-5x

+14x

-7

Write a heading using a common factor of 5x from the first row

5x

10x2

-5x

+14x

-7

Work out the headings for the rows, e.g. “What does 5x need to be multiplied by to make 10x2?”

2x

-1

5x

10x2

-5x

+14x

-7

Repeat for the heading for the remaining row, e.g. “What does 2x need to be multiplied by to make +14x?”

2x

-1

5x

10x2

-5x

+7

+14x

-7

Read off the brackets from the column and row headings

(2x - 1)(5x + 7)

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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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