GCSE Maths

Numbers
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Integers
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The Fibonacci Sequence
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Algebra
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Algebraic Notation
Algebraic Vocabulary
Collecting Like Terms
Expanding Brackets and Simplifying
Expanding Double Brackets
The Difference of Two Squares
Substituting into Formulae
Subject of Formula
Factorisation
Factorising Quadratic Expressions
More on Factorisation
Equations
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Map scales
Bearings
Area
Volume
Circles: arcs and sectors
Congruence and similarity
Pythagoras’ theorem
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Area of a triangle
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Factorising Quadratic Expressions

GCSE Maths Algebra Factorising Quadratic Expressions

Consider the expansion of (2x+1)(3x-4)

(2x+1)(3x-4)=2x(3x-4)+1(3x-4)

= 6x^{2}-8x+3x-4

= 6x^{2}-5x-4

So, if we have to factorise 6x^{2}-5x-4, we should obtain (2x+1)(3x-4).

When the coefficient of x^{2} is 1, it is easier to factorise the quadratic expression. We look for two numbers, their sum should be the coefficient of x and their product should be the constant term.

For example, to factorise x^{2}+5x+6, where the coefficient of x^{2} is 1, we seek two numbers, their sum = 5 and their product 6. These two numbers are 2 and 3. So, we should have:

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

We check by expanding (x+2)(x+3)

(x+2)(x+3)=x(x+3)+2(x+3)

= x^{2}+3x+2x+6

= x^{2}+5x+6

Let’s look at some examples.

Example

Factorise the following quadratic equations:

i) x^{2}+11x+30

ii) x^{2}-11x+30

iii) x^{2}-x-30

iv) x^{2}+x-30

v) x^{2}+30+11x

i) x^{2}+11x+30

Since 5+6=11 and 5 \times 6=30

x^{2}+11x+30=(x+5)(x+6)

 

ii) x^{2}-11x+30=(x-5)(x-6)

Since -5+-6=-11 and -5 \times-6=30

 

iii) x^{2}-x-30=(x+5)(x-6)

 

iv) x^{2}+x-30=(x-5)(x+6)

 

v) We should first re-write x^{2}+30+11x in the standard form: x^{2}+11x+30.

As seen earlier, this is (x+5)(x+6)

Example

Factorise -10-3x+x^{2}

-10-3x+x^{2}=x^{2}-3x-10

= (x+2)(x-5)

Example

Factorise x^{2}+8x+7

x^{2}+8x+7=(x+1)(x+7)

Example

Factorise x^{2}-x-110

x^{2}-x-110=(x+10)(x-11)

Example

Factorise x^{2}-11x+26

x^{2}-11x-26=(x+2)(x-13)

Example

Factorise x^{2}-9x+14

Since -2+-7=-9 and -2 \times-7=+14

x^{2}-9x+14=(x-2)(x-7)

Example

Factorise x^{2}-9x

x^{2}-9x=x(x-9)

Example

Factorise x^{2}-9

x^{2}-9=(x+3)(x-3)

Since 3-3=0 and 3 \times-3=-9

Example

Factorise the following:

i) x^{2}-16

ii) x^{2}-100

iii) x^{2}-1

iv) x^{2}-49

v) 2x^{2}-50

i) x^{2}-16=(x+4)(x-4)

 

ii) x^{2}-100=(x+10)(x-10)

 

iii) x^{2}-1=(x+1)(x-1)

 

iv) x^{2}-49=(x+7)(x-7)

 

v) 2x^{2}-50

First write 2x^{2}-50 as 2(x^{2}-25)

Then since x^{2}-25=(x+5)(x-5)

We have 2x^{2}-50=2(x^{2}-25)

= 2(x+5)(x-5)

Example

Factorise:

i) 3x^{2}-27

ii) 5x^{2}-20

i) 3x^{2}-27=3(x^{2}-9)

= 3(x+3)(x-3)

 

ii) 5x^{2}-20=5(x^{2}-4)

= 5(x+2)(x-2)

Example

Factorise:

i) x^{2}-121

ii) x^{2}y-121y

iii) 6x^{2}y-726y

i) x^{2}-121=(x+11)(x-11)

 

ii) x^{2}y-121y=y(x^{2}-121)

= y(x+11)(x-11)

 

iii) 6x^{2}y-726y=6y(x^{2}-121)

= 6y(x+11)(x-11)

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