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)