# Grouping Symbols: Expanding Brackets

> Multiply everything inside the brackets by the term outside, then tidy up.

From Selective School Maths on WhizzNotes, free study notes for NSW students: https://whizznotes.com/selective-school-maths/expanding-brackets/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: Party bags
Leah is making 3 party bags. Each bag gets $x$ lollies and 6 stickers.

How much is in all the bags? She could say 3 lots of $(x + 6)$ and write $3(x + 6)$. Or she could count each thing on its own: 3 bags of $x$ lollies is $3x$, and 3 bags of 6 stickers is 18 stickers. Altogether, $3x + 18$.

Both answers describe the same pile of stuff. Turning $3(x + 6)$ into $3x + 18$ is called **expanding the brackets**.

**In short:** Expanding brackets means multiplying the term outside by every term inside: $a(b + c) = ab + ac$. Keep each plus or minus sign, then collect any like terms.

## Before you start

- No sign next to a bracket means multiply: $3(x + 6) = 3 \times (x + 6)$.
- Multiplying terms: $4 \times 2x = 8x$ and $x \times x = x^2$. (See Multiplying and Dividing Terms.)
- Collecting like terms: $3x + 2x = 5x$. (See Adding and Subtracting Terms.)

## Brackets group things together

Brackets (also called **grouping symbols**) keep terms together as one package. The number or term outside multiplies the **whole** package.

### Key idea: Multiply every term inside
The term outside the brackets multiplies **each** term inside.

$$4(xy + 9) = 4 \times xy + 4 \times 9 = 4xy + 36$$

Draw two little arrows from the outside term, one to each term inside. Each arrow is one multiplication.

*Diagram: Area model: a rectangle 3 high, split into a part x wide and a part 6 wide, with areas 3x and 18*
*A rectangle 3 high and (x + 6) wide splits into two parts: 3x and 18.*

### Worked example: Numbers outside
$3(a + 6) = 3a + 18$

$5(5x - 4) = 25x - 20$

$4(3 + 2x) = 12 + 8x$

$7(8 - 5m) = 56 - 35m$

Keep the plus or minus sign that was inside. In $5(5x - 4)$, the 4 was being taken away, so $20$ is taken away too.

## A letter outside

The rule is exactly the same when a letter is outside.

### Worked example: Letters outside
$x(4 + y) = 4x + xy$

$a(5 + 2b) = 5a + 2ab$

$k(k + 3) = k^2 + 3k$, because $k \times k = k^2$

$2(2x^2 - x) = 4x^2 - 2x$

### Watch out: Do not stop after the first term
$3(x + 6)$ is **not** $3x + 6$. The 3 must multiply the 6 as well. Check with $x = 1$: $3(1 + 6) = 21$, and $3x + 18 = 21$. But $3x + 6 = 9$. Wrong!

## Expand, then collect

Often you expand two sets of brackets and then collect like terms.

### Worked example: Two brackets
$3(x + 2) + 2(x + 5)$

Expand: $3x + 6 + 2x + 10$

Collect: $5x + 16$

### Exam tip: Check with a number
Put $x = 1$ into the question and into your answer. $3(3) + 2(6) = 21$ and $5 + 16 = 21$. Same answer, so it is right.

### Challenge: Selective level: going backwards
**Factorising** is the reverse of expanding. Find the biggest number that divides every term and put it outside.

$12x + 18 = 6(2x + 3)$, because $6 \times 2x = 12x$ and $6 \times 3 = 18$.

A trickier one: $4(x + 3) - 2x$. Expand first: $4x + 12 - 2x = 2x + 12$.

Word problem: 5 friends each buy a pen for $p$ dollars and a pad for 3 dollars. Total cost $= 5(p + 3) = 5p + 15$ dollars.

**Mind map**

```text
Expanding brackets
  The rule
    Outside term times every inside term
    $4(xy + 9) = 4xy + 36$
  Number outside
    $5(5x - 4) = 25x - 20$
    Keep the sign
  Letter outside
    $k(k + 3) = k^2 + 3k$
  Then collect
    Expand both brackets
    Add like terms
  Check
    Put in $x = 1$
```

## Quick summary

- Brackets group terms. The term outside multiplies **every** term inside.
- $a(b + c) = ab + ac$.
- Keep the plus or minus sign from inside the brackets.
- A letter outside works the same way: $x(x + 2) = x^2 + 2x$.
- After expanding, collect like terms.
- Check by putting in a small number such as 1.

## Quiz

Try these questions from the chapter quiz.

1. $3(a + 6) = $
   - A. $3a + 6$
   - B. $3a + 18$
   - C. $9a$
   - D. $3a + 9$
   - Answer: B. $3a + 18$. Multiply both terms: $3 \times a = 3a$ and $3 \times 6 = 18$.

2. $5(5x - 4) = $
   - A. $25x - 4$
   - B. $10x - 9$
   - C. $25x - 20$
   - D. $25x + 20$
   - Answer: C. $25x - 20$. $5 \times 5x = 25x$ and $5 \times 4 = 20$, keeping the minus sign.

3. $9(8 + 3a) = $
   - A. $72 + 27a$
   - B. $72 + 3a$
   - C. $17 + 12a$
   - D. $99a$
   - Answer: A. $72 + 27a$. $9 \times 8 = 72$ and $9 \times 3a = 27a$.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/selective-school-maths/expanding-brackets/#quiz