Expanding and factorising

What you will learn

expand a product of two binomials using the distributive law,
recognise and apply the perfect-square expansion patterns,
recognise and apply the difference-of-squares pattern,
factorise monic quadratic trinomials of the form $x^2 + bx + c$ ,
understand expanding and factorising as inverse operations.

Worked example 0 Real-world example: paving a courtyard

A courtyard is $(x + 3)$ metres long and $(x + 2)$ metres wide. What is its area in expanded form?

Area $= (x+3)(x+2)$ .
Expand: $x \times x + x \times 2 + 3 \times x + 3 \times 2$ .
Simplify: $x^2 + 2x + 3x + 6 = x^2 + 5x + 6$ .
If $x = 10$ , the courtyard is $13 \times 12 = 156$ m $^2$ , and $10^2 + 5(10) + 6 = 156$ . It checks.

Key idea: expanding a binomial product gives a quadratic expression. The area model below shows why each term appears.

Area model: (x + 3)(x + 2) = x² + 5x + 6. Each region represents one term in the expansion.

1. Expanding binomial products

To expand $(x + a)(x + b)$ , multiply each term in the first bracket by each term in the second:

General binomial product

(x + a)(x + b) = x^2 + (a + b)x + ab.

The middle coefficient is the sum of $a$ and $b$ ; the constant term is the product of $a$ and $b$ .

Worked example 1 Basic binomial expansion

Expand $(x + 4)(x + 5)$ .

Using the formula: $a = 4$ , $b = 5$ .

(x+4)(x+5) = x^2 + (4+5)x + (4)(5) = x^2 + 9x + 20.

Worked example 2 With a negative term

Expand $(x - 3)(x + 7)$ .

Here $a = -3$ , $b = 7$ .

(x-3)(x+7) = x^2 + (-3+7)x + (-3)(7) = x^2 + 4x - 21.

Worked example 3 Both terms negative

Expand $(x - 6)(x - 2)$ .

Here $a = -6$ , $b = -2$ .

(x-6)(x-2) = x^2 + (-6 + (-2))x + (-6)(-2) = x^2 - 8x + 12.

2. Special products

Two patterns appear so often that they deserve their own formulas.

Special binomial products

Perfect square (sum)

(x + a)^2 = x^2 + 2ax + a^2.

Perfect square (difference)

(x - a)^2 = x^2 - 2ax + a^2.

Difference of squares

(x + a)(x - a) = x^2 - a^2.

Worked example 4 Perfect square expansion

Expand $(x + 7)^2$ .

(x+7)^2 = x^2 + 2(7)x + 7^2 = x^2 + 14x + 49.

Worked example 5 Difference of squares

Expand $(x + 9)(x - 9)$ .

(x+9)(x-9) = x^2 - 9^2 = x^2 - 81.

There is no middle term — the $+9x$ and $-9x$ cancel exactly.

3. Factorising monic quadratics

Factorising reverses expanding. Given $x^2 + bx + c$ , you need two numbers that add to $b$ and multiply to $c$ .

Factorising x² + bx + c

x^2 + bx + c = (x + p)(x + q) \quad \text{where } p + q = b \text{ and } pq = c.

Worked example 6 Factorising with two positive numbers

Factorise $x^2 + 7x + 12$ .

Find two numbers that add to $7$ and multiply to $12$ :

Try $3$ and $4$ : $3 + 4 = 7$ and $3 \times 4 = 12$ . Yes.

$x^2 + 7x + 12 = (x + 3)(x + 4).$

Check by expanding: $(x+3)(x+4) = x^2 + 4x + 3x + 12 = x^2 + 7x + 12$ . Correct.

Worked example 7 Factorising with a negative constant

Factorise $x^2 + 2x - 15$ .

Need two numbers that add to $2$ and multiply to $-15$ . Since the product is negative, one number is positive and one is negative:

Try $5$ and $-3$ : $5 + (-3) = 2$ and $5 \times (-3) = -15$ . Yes.

$x^2 + 2x - 15 = (x + 5)(x - 3).$

Worked example 8 Factorising with a negative middle term

Factorise $x^2 - 9x + 20$ .

Need two numbers that add to $-9$ and multiply to $20$ . Both must be negative (negative sum, positive product):

Try $-4$ and $-5$ : $(-4) + (-5) = -9$ and $(-4)(-5) = 20$ . Yes.

$x^2 - 9x + 20 = (x - 4)(x - 5).$

4. Factorising special forms

The special products from Section 2 can be reversed:

Factorising special forms

Difference of squares

x^2 - a^2 = (x + a)(x - a).

Perfect square trinomial

x^2 + 2ax + a^2 = (x + a)^2, \qquad x^2 - 2ax + a^2 = (x - a)^2.

Worked example 9 Factorising a difference of squares

Factorise $x^2 - 49$ .

Recognise $49 = 7^2$ , so this is a difference of squares:

$x^2 - 49 = (x + 7)(x - 7).$

Worked example 10 Factorising a perfect square trinomial

Factorise $x^2 - 10x + 25$ .

Check: is $25 = 5^2$ and is $10 = 2 \times 5$ ? Yes. So:

$x^2 - 10x + 25 = (x - 5)^2.$

Practice

Fluency

Tier 1: basic skills

Expand $(x + 2)(x + 6)$ .
Expand $(x + 5)(x - 3)$ .
Expand $(x - 4)(x - 7)$ .
Expand $(x + 8)^2$ .
Expand $(x - 3)^2$ .
Expand $(x + 11)(x - 11)$ .
Factorise $x^2 + 8x + 15$ .
Factorise $x^2 - 5x + 6$ .
Factorise $x^2 - 36$ .
Factorise $x^2 + 12x + 36$ .

Reasoning

Tier 2: mixed practice

Expand and simplify $(x + 3)(x + 4) - (x + 1)(x + 2)$ .
Factorise $x^2 + x - 20$ .
Factorise $x^2 - 3x - 28$ .
A square garden has side $(x + 5)$ m. A path of width $1$ m surrounds it. Find the area of the path in expanded form.
Factorise $x^2 - 100$ and hence evaluate $998^2 - 1000^2$ mentally.
Show that $(x + a)^2 - (x - a)^2 = 4ax$ by expanding both sides.
Factorise $x^2 - 14x + 49$ .
Factorise $2x^2 + 14x + 24$ completely.

Reasoning

Tier 3: explain and apply

Explain why $(x + a)(x - a)$ has no $x$ term. Use the area model or algebra to justify.
A rectangle has area $x^2 + 9x + 18$ cm $^2$ . Find expressions for its length and width.
Without expanding, decide whether $(x + 3)^2$ and $x^2 + 9$ are equivalent. Explain.
Factorise $x^2 - 2x - 35$ and use your factorisation to solve $x^2 - 2x - 35 = 0$ .
Explain the connection between expanding and factorising using the analogy of multiplication and division of numbers.

Challenge

Reasoning

Harder reasoning

Expand $(2x + 3)(x - 4)$ . (Note: this is a non-monic product — the coefficient of $x^2$ is not $1$ .)
Factorise $x^4 - 16$ completely. (Hint: treat $x^4$ as $(x^2)^2$ and apply difference of squares twice.)
Prove that the sum of any two consecutive odd numbers is divisible by $4$ . (Hint: let the odd numbers be $2n - 1$ and $2n + 1$ , and use difference of squares.)
If $x + \dfrac{1}{x} = 5$ , find the value of $x^2 + \dfrac{1}{x^2}$ . (Hint: square both sides and simplify.)