Factorising Simple Quadratics (Cambridge (CIE) IGCSE International Maths)

Revision Note

Written by: Jamie Wood

Reviewed by: Dan Finlay

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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 $x^{2} - 2 x - 8$

We need a pair of numbers that for $x^{2} + b x + c$
- 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:
  - $(x + 2) (x - 4)$

How do I factorise quadratics by grouping?

This is shown most easily through an example: factorising $x^{2} - 2 x - 8$

We need a pair of numbers that for $x^{2} + b x + c$
- 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
  - $x^{2} + 2 x - 4 x - 8$
- 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
  - $x (x + 2) - 4 (x + 2)$
- Note that these both now have a common factor of (x + 2) so this whole bracket can be factorised out
  - $(x + 2) (x - 4)$

How do I factorise quadratics using a grid?

This is shown most easily through an example: factorising $x^{2} - 2 x - 8$

We need a pair of numbers that for $x^{2} + b x + c$
- 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
- $(x + 2) (x - 4)$

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 a ≠ 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 $x^{2} - 4 x - 21$ .

We will factorise by inspection

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

$3 \times (- 7) = - 21 3 + (- 7) = - 4$

Write down the brackets

(x + 3)(x - 7)

(b) Factorise $x^{2} - 5 x + 6$ .

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

$(- 3) \times (- 2) = 6 (- 3) + (- 2) = - 5$

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)

We will factorise by using a grid

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

$4 \times (- 6) = - 24 4 + (- 6) = - 2$

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)

Last updated: 17 October 2024

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