More on Factorisation

We shall now factorise ax^{2}+bx+c, where the coefficient of x^{2} is not one. For example:

  • 12x^{2}+x-6
  • 4x^{2}-4x-3
  • 6x^{2}+5x-3

We shall find two numbers whose sum should be b and whose products should be ac. Once found, we break the middle term bx into the sum of these two numbers and factorise the resulting expression.

Consider 2x^{2}+5x-3, we shall find two numbers:

  • Sum = 5
  • Product = 2(-3) = -6

These two numbers are 6 and -1

We now rewrite 2x^{2}+5x-3 as 2x^{2} as 2x^{2}+6x-x-3

Now, 2x^{2}+6x-x-3=2x(x+3)-1(x+3), which is (2x-1)(x+3).

Even if we took 2x^{2}+5x-3 as 2x^{2}-x+6x-3, we shall have x(2x-1)+3(2x-1), which again factorises into (x+3)(2x-1), as before.

Let’s look at some examples.

Example

Factorise 12x^{2}+x-6

We need two numbers.

Sum = 1, product = 13(-6) = -72

These are 9 and 8.

So, we break x into 9x-8x, to obtain:

12x^{2}+x-6 = 12^x{2}+9x-8x-6

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

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


Example

Factorise 4x^{2}-4x-3

Sum = -4, product = -12

Since -6 \times 2 = -12 and -6 + 2 = -4, we have:

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

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

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


Example

Factorise 20x^{2}+3x-9

20x^{2}+3x-9=20x^{2}-12x+15x-9

= 4x(5x-3)+3(5x-3)

= (4x+3)(5x-3)


Example

Factorise 14x^{2}-17x-6

14x^{2}-7x-6=14x^{2}+4x-21x-6

= 2x(7x+2)-3(7x+2)

= (2x-3)(7x+2)


Example

Factorise 6x^{2}-6-5x

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

= 6x^{2}-9x+4x-6

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

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


Example

Factorise 4x^{2}-4x-3

Sum =-4, product =-12

Since -6 \times 2 = -12 and -6+2 = -4 , we have:

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

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

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


Example

Factorise 20x^{2}+3x-9

20x^{2}+3x-9=20x^{2}-12x+15x-9

= 4x(5x-3)+3(5x-3)

= (4x+3)(5x-3)


Example

14x^{2}-17x-6

14x^{2}-17x-6=14x^{2}+4x-21x-6

= 2x(7x+2)-3(7x+2)

= (2x-3)(7x+2)


Example

Factorise 6x^{2}-5x-6

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

= 6x^{2}-9x+4x-6

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

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


Example

Factorise 8x^{2}-22x+15

We look for two numbers with a sum of -22 and a product of 8\times15 = 120.

With trial and error, we find -10 and -12. So, we have:

8x^{2}-22x+15=8x^{2}-10x-12x+15

= 2x(4x-5)-3(4x-5)

= (2x-3)(4x-5)


Example

Factorise 12x^{2}+5x-3

12x^{2}+5x-3=12x^{2}+9x-4x-3

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

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