Factorising Quadratic Expressions

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:

Factorise the following quadratic expressions:

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)