### GCSE Maths

Numbers
Algebra
Geometry and Measures
Probability
Statistics

Factorising quadratic expressions involves rewriting them as the product of simpler factors, which makes them easier to manipulate or solve. Quadratic expressions take the form of ax² + bx + c, where a, b, and c are constants.

When the number in front of x² (the coefficient) is 1, factorising the quadratic expression is much simpler. For example, consider the quadratic expression x² + 7x + 12. Here, the coefficient of x² is 1.

To factorise this expression, we need to find two numbers that:

• Add up to the coefficient of the x term (7 in this case)
• Multiply to get the constant term (12 in this case)

After trying out a few possibilities, we find that the numbers 3 and 4 fit these criteria:

• Sum: 3 + 4 = 7
• Product: 3 × 4 = 12

Now, we can factorise the expression:

x² + 7x + 12 = (x + 3)(x + 4)

Let’s look at some examples.

## Examples

Example 1:

i) x² + 11x + 30

ii) x² – 11x + 30

iii) x² – x – 30

iv) x² + x – 30

i) x² + 11x + 30 = (x + 5)(x + 6)

ii) x² – 11x + 30 = (x – 5)(x – 6)

iii) x² – x – 30 = (x + 5)(x – 6)

iv) x² + x – 30 = (x – 5)(x + 6)

Example 2:

Factorise the following:

A) -10 – 3x + x²

B) x² + 8x + 7

C) x² – x – 110

D) x² – 11x + 26:

E) x² – 9x + 14:

F) x² – 9x:

G) x² – 9:

A) -10 – 3x + x² = x² – 3x – 10

= (x + 2)(x – 5)

B) x² + 8x + 7 = (x + 1)(x + 7)

C) x² – x – 110 = (x + 10)(x – 11)

D) x² – 11x + 26 = (x – 2)(x – 13)

E) x² – 9x + 14 = (x – 2)(x – 7)

F) x² – 9x = x(x – 9)

G) x² – 9 = (x + 3)(x – 3)

Example 3:

Factorise the following:

i) x² – 16

ii) x² – 100

iii) x² – 1

iv) x² – 49

i) x² – 16 = (x + 4)(x – 4)

ii) x² – 100 = (x + 10)(x – 10)

iii) x² – 1 = (x + 1)(x – 1)

iv) x² – 49 = (x + 7)(x – 7)