Back to GCSE Maths Revision Home

Factorising Quadratics

GCSELevel 4-5Level 6-7AQAEdexcelEdexcel iGCSEOCRWJEC

Factorising Quadratics Revision

Factorising Quadratics

Quadratics are algebraic expressions that include the term, x^2, in the general form,

ax^2 + bx + c

Where a, b, and c are all numbers. We’ve seen already seen factorising into single brackets, but this time we will be factorising quadratics into double brackets.

(nx+m)(px+q)

There are 2 main types of quadratics you will need to be able to factorise; one where a=1 and the other where a\neq1.

Make sure you are happy with the following topics before continuing.

Level 4-5GCSEAQAEdexcelOCRWJECEdexcel iGCSE

Take Note: The Factorising Trick

There is a quick trick to determine whether you should add a + or - sign to your brackets. There are three sub-types which we will go over here.

Sub-type (a): contains all positives

These quadratics contain all positive terms, e.g. x^2 +5x + 6. When factorised, both brackets will contain \bf{\bf{\large{+}}}.

x^2 + 5x + 6 = (x+3)(x+2)

Sub-type (b): b is negative and c is positive

These quadratics contain a negative b value and a positive c value. When factorised, both brackets will contain \bf{\large{-}}.

x^2 -10x +21 = (x-7)(x-3)

Sub-type (c): c is negative.

If c is negative, when factorised, one bracket will contain a \bf{+} the other will contains a \bf{\large{-}}. The order or these will need to be determined. These are the hardest type and require the most thought.

x^2 + 3x -18 = (x+6)(x-3)

MME Logo
TikTok

Your 2024 Revision Partner

@mmerevise

Open TikTok
Level 4-5GCSEAQAEdexcelOCRWJECEdexcel iGCSE

Type 1: Factorising quadratics (a=1)

When we say a=1, we mean the number before x^2 in  x^2+bx+c will be 1 (typically we don’t write the 1). Any number that appears before an x term is called a coefficient, so in this case, a is the coefficient of x^2 which has a value of 1.

Example: Factorise the following quadratic into two brackets, x^2\textcolor{blue}{-3}x+\textcolor{red}{2}

Step 1: First we can write two brackets with an x placed in each bracket.

(x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,) (x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,)

Step 2: We can identify that this is a sub-type (b) quadratic, meaning both brackets will contain \large{-}

(x\,\,\,\,\,\,\,\,-\,\,\,\,\,\,\,\,) (x\,\,\,\,\,\,\,-\,\,\,\,\,\,\,\,)

Step 3: We have to find two numbers which multiply to make \textcolor{red}{2} and when added together make \textcolor{blue}{-3}.

We know both numbers will be negative.

-2 \times -1 = \textcolor{red}{2}

-2 + -1 = \textcolor{blue}{-3}

Finally add these numbers to the brackets.

(x-2) (x-1)

Level 4-5GCSEAQAEdexcelOCRWJECEdexcel iGCSE
Level 6-7GCSEAQAEdexcelOCRWJECEdexcel iGCSE

Type 2: Factorising quadratics (a> 1)

In this instance the general form of the equation is ax^2+bx+c where a>1.

Example: Factorise the following quadratic 4x^2+\textcolor{blue}{3x}\textcolor{red}{-1}

Step 1: When a>1 it makes things more complicated. It is not immediately obvious what the coefficient of each x term should be. There are two possible options,

(4x \kern{1 cm} ) (x  \kern{1 cm} ) or (2x \kern{1 cm} ) (2x  \kern{1 cm} )

Step 2: We can identify that this quadratic is part of sub-type (c) meaning it can contain + and -

This is most important for quadratic pairs which are non-symmetrically creating a third option, all three are shown below.

\begin{aligned}(4x \kern{0.4 cm} +\kern{0.4 cm} )&(x  \kern{0.4 cm}-\kern{0.4 cm} )\\ (4x \kern{0.4 cm} -\kern{0.4 cm} )&(x  \kern{0.4 cm}+\kern{0.4 cm} )\\(2x \kern{0.4 cm}+\kern{0.4 cm} )&(2x \kern{0.4 cm}-\kern{0.4 cm} )\end{aligned}

Step 3: We need to find two numbers which when multiplied make \textcolor{red}{-1}

\textcolor{red}{-1} has only one factor.

-1 \times 1 = -1

Step 4: We need to find a combination which gives \textcolor{blue}{3x}

We can test our 3 possibilities,

\begin{aligned}(4x+1)(x -1) &= 4x^2 -3x -1 \\(4x-1)(x+1) &= 4x^2 + 3x -1 \\(2x+1)(2x-1) &= 4x^2 -1\end{aligned}

As we can see (4x-1) (x+1) gives the correct expansion and is therefore the answer.

Level 6-7GCSEAQAEdexcelOCRWJECEdexcel iGCSE

Example 1: Factorising Simple Quadratics

Factorise x^2 - x - 12.

[2 marks]

Step 1: Draw empty brackets

(x \kern{1 cm} ) (x  \kern{1 cm} )

Step 2: Identify sub-type (b)

(x \kern{0.4cm} + \kern{0.4cm} ) (x  \kern{0.4 cm} - \kern{0.4cm} )

Step 3: We are looking for two numbers which multiply to make \textcolor{red}{-12} and add to make \textcolor{blue}{-1}. Let’s consider some factor pairs of -12.

\begin{aligned}(-1)\times12&=-12  \,\,\text{    and  } -1 + 12 = 11\\(-2)\times6&=-12 \,\,\text{    and  } -2 + 6 = 4\\ (-6)\times2&=-12 \,\,\text{  and  } -6 + 2 = -4\\ (-3)\times4&=-12 \,\,\text{    and  } -3 +4 = 1\\ \textcolor{red}{(-4)\times3}&\textcolor{red}{=-12 }\,\,\text{    and  } \textcolor{blue}{-4 + 3 = -1}\end{aligned}

We could keep going, but there’s no need because the last pair, -4 and 3, add to make -1. This pair fills both criteria, (as highlighted above) so the factorisation of x^2 - x - 12 is

(x - 4)(x + 3)

Note: You can try expanding the double brackets to check your answer is correct. You should always get your original quadratic equation if you do this correctly.

Level 4-5GCSEAQAEdexcelOCRWJECEdexcel iGCSE

Example 2: Factorising Harder Quadratics 

Factorise 2x^2 + 7x + 3.

[3 marks]

Step 1: Draw the empty brackets. Even though a>1 there is only one possible option this time.

(2x \kern{1 cm} ) (x  \kern{1 cm} )

Step 2: Identify sub-type (a), meaning both brackets contain +.

(2x \kern{0.4cm} + \kern{0.4cm} ) (x  \kern{0.4 cm} + \kern{0.4cm} )

 

Step 3: Find two numbers which multiply to give \textcolor{red}{3}

3 only has one factor.

3 \times 1  = 3

Step 4: Find the combination which gives 7x

\begin{aligned}\textcolor{blue}{(2x \,\, +\,\, 1) (x\,\, + \,\,3) }&\textcolor{blue}{= 2x^2 + 7x +3} \\ (2x \,\, +\,\, 3) (x\,\, + \,\,1) &= 2x^2 +5x + 3\end{aligned}

As we can see, (2x + 1)(x + 3), gives the correct expansion and is therefore the correct answer.

Level 6-7GCSEAQAEdexcelOCRWJECEdexcel iGCSE

Factorising Quadratics Example Questions

We are looking for two numbers which add to make 1 and multiply to make -30.

 

The factors of 30 that satisfy theses two requirements are 5 and 6.

 

Therefore, the full factorisation of a^2 + a - 30 is

 

(a - 5)(a + 6)

MME Premium Laptop

Save your answers with

MME Premium

Gold Standard Education

We are looking for two numbers which add to make -5 and multiply to make 6.

 

The factors of 6 that satisfy theses two requirements are -2 and -3.

 

Therefore, the full factorisation of k^2 - 5k + 6 is

 

(k - 2)(k - 3)

MME Premium Laptop

Save your answers with

MME Premium

Gold Standard Education

We are looking for two numbers which add to make 7 and multiply to make 12.

 

The factors of 12 that satisfy theses two requirements are 3 and 4.

 

Therefore, the full factorisation of x^2 + 7x + 12 is,

 

(x + 3)(x + 4)

MME Premium Laptop

Save your answers with

MME Premium

Gold Standard Education

In the quadratic, a = 3, 11 is positive and c is positive. We can set up the brackets as follows: (3x \,\,\, + \,\,\,)(x \,\,\, + \,\,\,).

We are looking for two positive numbers which multiply to make 6. The possible factors of 6 are

\begin{aligned}(6)\times(1)&=6 \\ (3)\times(2)&=6 \end{aligned}

We now test all the combinations:

\begin{aligned}(3x+6)(x+1)&=3x^2+6x+3x+6 \\ (3x+1)(x+6)&= 3x^2+x+18x+6 \\ (3x+3)(x+2)&= 3x^2+6x+3x+6 \\ (3x+2)(x+3) &= 3x^2+9x+2x+6\end{aligned}

Hence the correct factorisation is (3x+2)(x+3)

MME Premium Laptop

Save your answers with

MME Premium

Gold Standard Education

We can see this is a sub-type (c) meaning it will contain both + and -

Factors of -6

(-1) \times 6 = -6

(-6) \times 1 = -6

(-2) \times 3 = -6

(-3) \times 2 = -6

Lets find the options which give - 5m

(2m+2)(2m-3) = 4m^2 - 2m - 6

(4m + 1)(m-6) = 4m^2 -23m-6

(4m-1)(m+6) = 4m^2 +23m -6

(4m+3)(m-2) = 4m^2 -5m -6

We can see that last option with +3 and -2 is the correct combination.

This gives the final answer to be:

(4m+3)(m-2)

MME Premium Laptop

Save your answers with

MME Premium

Gold Standard Education

Factorising Quadratics Worksheet and Example Questions

Site Logo

(NEW) Factorising Quadratics Exam Style Solutions - MME

Level 4-5GCSENewOfficial MME
Site Logo

(NEW) Factorising Harder Quadratics Exam Style Questions - MME

Level 6-7GCSENewOfficial MME

Factorising Quadratics Drill Questions

Site Logo

Factorising Quadratics - Drill Questions

Level 4-5GCSE
Site Logo

Algebra Quadratics Expand and Factorise - Drill Questions

Level 4-5GCSE
MME Premium UI
Product

MME Premium Membership

£19.99

/month

Learn an entire GCSE course for maths, English and science on the most comprehensive online learning platform. With revision explainer videos & notes, practice questions, topic tests and full mock exams for each topic on every course, it’s easy to Learn and Revise with the MME Learning Portal.

Sign Up Now

Related Topics

MME

Collecting Like Terms – Revision and Worksheets

Level 1-3GCSEKS3
MME

Powers and Roots Worksheets, Questions and Revision

Level 4-5GCSEKS3
MME

Expanding Brackets Worksheet, Questions and Revision

Level 1-3GCSEKS3
MME

Factorising | Questions, Worksheets and Revision

Level 4-5GCSEKS3