YEAR 9 · ALGEBRA · 15 MIN

Expanding single brackets

Use distribution to remove one bracket, including negative multipliers and subtracted groups.

In this lesson

Distribute to every bracket term.
Handle negative multipliers.
Connect expansion with an area model.

Before you start

Multiplying terms
Collecting like terms

01 / LEARN

Understand the idea.

Multiply the whole group

The distributive law multiplies every term in a group: $a (b + c) = a b + a c$ . For a rectangle with height $3$ and width $x + 4$ , the areas $3 x$ and $12$ join to make the total $3 (x + 4)$ .

Write one multiplication for each term inside the bracket before collecting anything. This separates expansion from simplification and makes an omitted product easier to spot. An expression outside the bracket stays separate unless it is the multiplier.

Distribute signs as well as numbers

An outside negative multiplier acts on the signed terms inside the bracket. For $- 2 (3 x - 5)$ , multiply both $3 x$ and $- 5$ by $- 2$ . The second product is positive because a negative multiplied by a negative is positive.

A minus immediately before a bracket is multiplication by $- 1$ . Thus subtracting $2 x - 3$ means adding $- 2 x + 3$ . Expand each complete group, preserving its signs, and only then collect like terms.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Expand and collect

Simplify $4 (2 x + 3) - 5 x$ .

Worked solution

1
Distribute 4 to both bracket terms.
$4 (2 x + 3) - 5 x = 4 (2 x) + 4 (3) - 5 x$
Both 2x and 3 belong to the group multiplied by 4.
2
Evaluate the products.
$8 x + 12 - 5 x$
The term -5x was outside the bracket and remains unchanged.
3
Collect the x terms.
$(8 - 5) x + 12 = 3 x + 12$
The constant 12 cannot combine with an x term.

Answer

$3 x + 12$

EXAMPLE 02

Subtract a whole product

Simplify $6 - 3 (2 y - 4)$ .

Worked solution

1
Treat -3 as the bracket multiplier.
$6 - 3 (2 y - 4) = 6 + (- 3) (2 y) + (- 3) (- 4)$
The subtraction sign is included in the multiplier of both terms.
2
Calculate the signed products.
$6 - 6 y + 12$
A negative times a positive is negative; a negative times a negative is positive.
3
Collect constants.
$(6 + 12) - 6 y = 18 - 6 y$
The two constants combine while the y term remains separate.

Answer

$18 - 6 y$

03 / YOUR TURN

Try it yourself.

Check your understanding

Expand $- 4 (x - 3)$ .

Show answer & reasoning

$- 4 x + 12$

$- 4 (x - 3) = (- 4) x + (- 4) (- 3) = - 4 x + 12$ . Both terms receive the negative multiplier.

Keep practising

6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

Expand $3 (x + 7)$ .
2 marks
Expand $5 (2 a - 3)$ .
2 marks
Expand $- 2 (4 p + 1)$ .
2 marks
Simplify $7 - (3 x + 2)$ .
2 marks
Simplify $2 (3 y - 4) + 5 (y + 1)$ .
3 marks
A rectangle is 6 cm high and $(x + 2)$ cm wide. Write its area without brackets.
3 marks

Read the worked solutions

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

Expand $3 (x + 7)$ .
2 marks
$3 x + 3 (7) = 3 x + 21$ .
$3 x + 21$
Expand $5 (2 a - 3)$ .
2 marks
Multiply both terms by 5.
$10 a - 15$
Expand $- 2 (4 p + 1)$ .
2 marks
$(- 2) (4 p) + (- 2) (1) = - 8 p - 2$ .
$- 8 p - 2$
Simplify $7 - (3 x + 2)$ .
2 marks
$7 - 3 x - 2 = 5 - 3 x$ .
$5 - 3 x$
Simplify $2 (3 y - 4) + 5 (y + 1)$ .
3 marks
Expand: $6 y - 8 + 5 y + 5$ .
Collect: $11 y - 3$ .
$11 y - 3$
A rectangle is 6 cm high and $(x + 2)$ cm wide. Write its area without brackets.
3 marks
$A = 6 (x + 2) = 6 x + 12$ cm $2$ .
$6 x + 12$ cm $2$

Checking saved progress…

Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum

Year 9 algebraic simplification and equivalent expressions, with consolidation of linear formula skills. Binomial expansion and quadratic factorisation/equations are in the separate Quadratics and Parabolas chapter.

Official curriculum source ↗

In this lesson

Understand the idea.

Multiply the whole group

Distribute signs as well as numbers

See the method in action.

Expand and collect

Distribute 4 to both bracket terms.

Evaluate the products.

Collect the x terms.

Subtract a whole product

Treat -3 as the bracket multiplier.

Calculate the signed products.

Collect constants.

Try it yourself.

Check your understanding

Keep practising