Expanding a single bracket | Year 8

01 / LEARN

The idea, step by step

The factor reaches every term

Expansion turns a product with a bracket into an equivalent sum: $a (b + c) = a b + a c$ . It works because $a$ copies of a sum contain $a$ copies of each part. For $4 (x + 3)$ , multiply $x$ and $3$ by $4$ to get $4 x + 12$ .

The new expression is equal to the old expression for every input. Expansion does not solve for a variable; it changes the form of an expression.

Distribute signs, then collect

In $- 2 (3 x - 5)$ , the factors are $- 2$ and each signed term. Their products are $- 6 x$ and $+ 10$ . The second product is positive because a negative times a negative is positive.

After expansion, collect any like terms outside the bracket. A numerical check can detect many mistakes, while the distributive steps justify the identity for every value.

A common trap

Writing $5 (x + 2) = 5 x + 2$ .

Instead: The factor multiplies both terms, so $5 (x + 2) = 5 x + 10$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

Expand then simplify

Simplify $3 (x + 4) + 2 x$ .

Expand to $3 x + 12 + 2 x$ .
Collect $3 x + 2 x = 5 x$ .

5 x + 12

Example 2

Negative outside factor

Expand $- 3 (2 y - 5)$ .

$(- 3) (2 y) = - 6 y$ .
$(- 3) (- 5) = 15$ .
Add the two products.

- 6 y + 15

Check your understanding

Expand $- 2 (x - 6)$ .

Show answer

$- 2 x + 12$

Multiply $x$ and $- 6$ by $- 2$ .

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Expand $4 (x + 2)$ .
2 marks
Expand $3 (2 a - 5)$ .
2 marks
Expand $- 5 (y + 3)$ .
2 marks
Expand $- 2 (4 p - 7)$ .
2 marks
Simplify $2 (x + 3) + 5 x$ .
2 marks
A strip is $3$ cm wide and $(n + 4)$ cm long. Write its area in expanded form.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Expand $4 (x + 2)$ .
2 marks
Multiply both terms by $4$ .
$4 x + 8$
Expand $3 (2 a - 5)$ .
2 marks
$3 (2 a) = 6 a$ and $3 (- 5) = - 15$ .
$6 a - 15$
Expand $- 5 (y + 3)$ .
2 marks
Distribute $- 5$ to both terms.
$- 5 y - 15$
Expand $- 2 (4 p - 7)$ .
2 marks
$(- 2) (4 p) = - 8 p$ and $(- 2) (- 7) = 14$ .
$- 8 p + 14$
Simplify $2 (x + 3) + 5 x$ .
2 marks
Expand to $2 x + 6 + 5 x$ .
Collect the $x$ terms.
$7 x + 6$
A strip is $3$ cm wide and $(n + 4)$ cm long. Write its area in expanded form.
3 marks
Area is $3 (n + 4)$ .
Distribute to get $3 n + 12$ .
$(3 n + 12)$ square centimetres

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Curriculum reference & lesson scope

Australian Curriculum v9 Mathematics with SA curriculum context

Combining like terms; substitution with signed/rational values; multiplying/dividing algebraic terms; expanding a single bracket; factorising common factors; equivalent linear expressions; rearranging simple formulas.

Official curriculum source ↗

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