Lesson 16.3 · 30 min

Factorising Quadratic Expressions

Reversing expansion: finding two numbers that multiply to the constant and add to the $x$ coefficient.

In this lesson you will learn to…

Factorise $x^{2} + b x + c$ into two brackets
Handle negative constants and coefficients
Factorise the difference of two squares
Check a factorisation by expanding

Why this matters

Factorising is how you solve most quadratic equations by hand — in the next lesson a factorised quadratic will give you its roots in one step.

Always check

Expand your answer: $(x + 3)(x - 6) = x^2 - 6x + 3x - 18 = x^2 - 3x - 18$ ✓. It takes ten seconds and catches most sign errors.

When there is no $x$ term

$x^2 + 0x - 49$ has $b = 0$ , so the two numbers are equal and opposite: $7$ and $-7$ . That is the difference of two squares.

Key rules and formulas

Product and sum
$x^{2} + b x + c = (x + p) (x + q) where p \times q = c, p + q = b$
For $x^{2} + 7 x + 12$ : which factor pair of 12 adds to 7? $3 \times 4 = 12$ and $3 + 4 = 7$ , so $(x + 3) (x + 4)$ .
Signs
$c$ positive, $b$ positive → both numbers positive. $c$ positive, $b$ negative → both negative. $c$ negative → one of each, the bigger one taking the sign of $b$ .
Difference of two squares
$x^{2} - k^{2} = (x + k) (x - k)$
$x^{2} - 49 = (x + 7) (x - 7)$ and $81 - 4 x^{2} = (9 + 2 x) (9 - 2 x)$ .

Worked examples

Worked example 1

All positive

Factorise

x^{2} + 10 x + 24

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

A negative constant

Factorise

x^{2} - 3 x - 18

Try it yourself first, then reveal the solution one step at a time.

Worked example 3

Two squares

Factorise

81 - 4 x^{2}

Try it yourself first, then reveal the solution one step at a time.

Practice: factorising

9 questions

Nine questions. Type factorised answers with brackets, e.g. (x+3)(x-6). Your answer must be in factorised form.

Question 1 of 9Fill the gapsFoundation
Complete each factorisation.
$x^{2} + 7 x + 6 = (x + 6) (x + □)$ : Gap a $x^{2} + 8 x + 15 = (x + 5) (x + □)$ : Gap b $x^{2} - 6 x + 8 = (x - 2) (x - □)$ : Gap c
Question 2 of 9Algebraic answer
Factorise $x^{2} + 7 x + 12$ .
Your expression
Type powers with ^ (so n^2 is n squared). Any equivalent form is accepted, but keep it in the form the question asks for.
Question 3 of 9Algebraic answer
Factorise $x^{2} - 3 x - 18$ .
Your expression
Type powers with ^ (so n^2 is n squared). Any equivalent form is accepted, but keep it in the form the question asks for.
Question 4 of 9Algebraic answer
Factorise $x^{2} + 2 x - 15$ .
Your expression
Type powers with ^ (so n^2 is n squared). Any equivalent form is accepted, but keep it in the form the question asks for.
Question 5 of 9Algebraic answer
Factorise $x^{2} - 9 x + 8$ .
Your expression
Type powers with ^ (so n^2 is n squared). Any equivalent form is accepted, but keep it in the form the question asks for.
Question 6 of 9Algebraic answer
Factorise $x^{2} - 49$ .
Your expression
Type powers with ^ (so n^2 is n squared). Any equivalent form is accepted, but keep it in the form the question asks for.
Question 7 of 9Algebraic answer
Factorise $81 - 4 x^{2}$ .
Your expression
Type powers with ^ (so n^2 is n squared). Any equivalent form is accepted, but keep it in the form the question asks for.
Question 8 of 9Quick-fire drill
Quick-fire: find the two numbers that multiply to the first and add to the second. Type them like 3, 4.
6 questions · 90 seconds · answer as many as you can.
Question 9 of 9Multiple choice
Which is the factorisation of $x^{2} + 4 x - 21$ ?

In this lesson you will learn to…

Why this matters

Always check

When there is no xxx term

Key rules and formulas

Product and sum

Signs

Difference of two squares

Worked examples

Practice: factorising

When there is no $x$ term