Expanding Linear Expressions with Brackets

Master the distributive law to multiply out brackets, handle positive and negative multipliers accurately, expand nested brackets, and simplify algebraic expressions.

Before you start

Multiplying in parts: $3 \times 12 = 3 \times 10 + 3 \times 2 = 36$ .
Rules for multiplying directed numbers: $(+) \times (+) = (+)$ , $(+) \times (-) = (-)$ , $(-) \times (+) = (-)$ , and $(-) \times (-) = (+)$ .
Combining like terms: Only terms with identical variable parts can be combined, for example $8a - 3a = 5a$ , while $5a - 10$ cannot be simplified further.
Order of operations: multiplication comes before addition and subtraction, so in $10 - 2 \times 3$ work out $2 \times 3$ first.

1. The Distributive Law

In algebra, expanding an expression (also called multiplying out) means removing brackets by multiplying every term inside the bracket by the term or number outside.

The rule that governs this is called the distributive law of multiplication over addition and subtraction.

Imagine a large rectangle of height $a$ divided into two smaller rectangles side by side with lengths $b$ and $c$ :

- The total length of the base is $b + c$ , so the area of the large rectangle is $a(b + c)$ .

- The areas of the two smaller rectangles are $ab$ and $ac$ .

- Because the whole area equals the sum of its parts, we have:

$a(b + c) = ab + ac$

When subtracting, the same principle gives $a(b - c) = ab - ac$ .

The Distributive Law

For any numbers or algebraic terms $a, b,$ and $c$ :

$a(b + c) = ab + ac$

$a(b - c) = ab - ac$

Every single term inside the bracket must be multiplied by the multiplier outside.

Expression	Multiplication Steps	Expanded Result
$2(x + 5)$	$2 \times x + 2 \times 5$	$2x + 10$
$4(3y - 2)$	$4 \times 3y - 4 \times 2$	$12y - 8$
$3(2a + 4b - 5c)$	$3 \times 2a + 3 \times 4b - 3 \times 5c$	$6a + 12b - 15c$
$5(4m - 1)$	$5 \times 4m - 5 \times 1$	$20m - 5$

Worked Example 1: Expanding Single Brackets with Positive Numbers

Expand each of the following expressions completely:
(a) $3(4x - 5)$
(b) $5(2a - 3b + 7)$

Solution:

(a) Multiply each term inside $(4x - 5)$ by $3$ :

$3(4x - 5) = (3 \times 4x) - (3 \times 5) = 12x - 15$

(b) Multiply each of the three terms inside $(2a - 3b + 7)$ by $5$ :

$\begin{aligned} 5(2a - 3b + 7) &= (5 \times 2a) - (5 \times 3b) + (5 \times 7) \\ &= 10a - 15b + 35 \end{aligned}$

(a) $12x - 15$ (b) $10a - 15b + 35$

2. Negative Multipliers and the Invisible One

When the multiplier outside the bracket is negative, extra care must be taken with the signs. Every term inside the bracket is multiplied by a negative number, which reverses each term’s sign.

Recall the fundamental rules of signs for multiplication:

- A negative times a positive gives a negative: $(-a) \times b = -ab$

- A negative times a negative gives a positive: $(-a) \times (-c) = +ac$

Expanding with Negative Multipliers

$-a(b + c) = -ab - ac$

$-a(b - c) = -ab + ac$

Notice that in $-a(b - c)$ , the second term becomes $+ac$ because $(-a) \times (-c) = +ac$ .

The Minus Sign Directly in Front of a Bracket

When an expression has a minus sign directly outside a bracket with no visible number, such as $-(2x - 5)$ , there is an invisible $1$ multiplying the bracket:

$-(2x - 5) = -1(2x - 5) = (-1) \times 2x + (-1) \times (-5) = -2x + 5$

A leading minus sign changes the sign of every term inside the bracket.

Worked Example 2: Negative Multipliers and Single Minus Signs

Expand each of the following completely:
(a) $-4(2x - 3)$
(b) $-(3a - 7b + 2)$
(c) $-2(-4m + 5n - 6)$

Solution:

(a) Multiply each term by $-4$ , noting $(-4) \times (-3) = +12$ :

$\begin{aligned} -4(2x - 3) &= (-4) \times 2x + (-4) \times (-3) \\ &= -8x + 12 \end{aligned}$

(b) Treat the leading minus sign as $-1$ multiplying each term inside:

$\begin{aligned} -(3a - 7b + 2) &= -1(3a - 7b + 2) \\ &= -3a + 7b - 2 \end{aligned}$

(c) Multiply each term by $-2$ , carefully tracking negative signs:

$\begin{aligned} -2(-4m + 5n - 6) &= (-2) \times (-4m) + (-2) \times 5n + (-2) \times (-6) \\ &= 8m - 10n + 12 \end{aligned}$

(a) $-8x + 12$ (b) $-3a + 7b - 2$ (c) $8m - 10n + 12$

3. Expanding and Collecting Like Terms

Often you must expand one or more pairs of brackets and then combine like terms to write the expression in its simplest form.

The 3-Step Strategy for Expanding and Simplifying

Step 1: Expand brackets completely. Multiply out every set of brackets using the distributive law. Pay close attention to negative multipliers.

Step 2: Group like terms. Rearrange terms so that like terms are adjacent, keeping the sign ( $+$ or $-$ ) attached to each term.

Step 3: Combine like terms. Add or subtract the numerical coefficients of like terms. Write the final answer with variables in alphabetical order and constants last.

Worked Example 3: Expanding Two Sets of Brackets and Combining Like Terms

Expand and simplify each expression completely:
(a) $3(2x + 1) + 4(x - 3)$
(b) $4(2a - 1) - 3(a + 2)$
(c) $5(2x - 3y) - 2(x - 4y) + 3x$

Solution:

(a) Expand both brackets, then group like terms:

$\begin{aligned} 3(2x + 1) + 4(x - 3) &= 6x + 3 + 4x - 12 \\ &= (6x + 4x) + (3 - 12) \\ &= 10x - 9 \end{aligned}$

(b) Expand both brackets. The first has multiplier $4$ , the second has multiplier $-3$ :

$\begin{aligned} 4(2a - 1) - 3(a + 2) &= 8a - 4 - 3a - 6 \\ &= (8a - 3a) + (-4 - 6) \\ &= 5a - 10 \end{aligned}$

(c) Expand both brackets, then collect $x$ -terms and $y$ -terms:

$\begin{aligned} 5(2x - 3y) - 2(x - 4y) + 3x &= 10x - 15y - 2x + 8y + 3x \\ &= (10x - 2x + 3x) + (-15y + 8y) \\ &= 11x - 7y \end{aligned}$

(a) $10x - 9$ (b) $5a - 10$ (c) $11x - 7y$

Beware the Order of Operations Trap!

In an expression like $10 - 2(3x - 1)$ , students are often tempted to calculate $10 - 2 = 8$ first. This is incorrect!

According to the order of operations (BODMAS), multiplication takes precedence over subtraction.

The $2$ is attached to $(3x - 1)$ by multiplication. Treat $-2$ as the multiplier:

$10 - 2(3x - 1) = 10 - 6x + 2 = -6x + 12$

Worked Example 4: Order of Operations with Outside Terms

Expand and simplify each of the following:
(a) $4x - 2(3x - 5)$
(b) $15 - 3(2a + 4) + 5a$
(c) $8 - (5y - 3) + 2y$

Solution:

(a) Multiply $(3x - 5)$ by $-2$ first, then combine with $4x$ :

$\begin{aligned} 4x - 2(3x - 5) &= 4x - 6x + 10 \\ &= -2x + 10 \end{aligned}$

(b) Expand the bracket with multiplier $-3$ before doing subtraction with $15$ :

$\begin{aligned} 15 - 3(2a + 4) + 5a &= 15 - 6a - 12 + 5a \\ &= (-6a + 5a) + (15 - 12) \\ &= -a + 3 \end{aligned}$

(c) Treat the minus sign in front of $(5y - 3)$ as $-1$ :

$\begin{aligned} 8 - (5y - 3) + 2y &= 8 - 5y + 3 + 2y \\ &= (-5y + 2y) + (8 + 3) \\ &= -3y + 11 \end{aligned}$

(a) $-2x + 10$ (b) $-a + 3$ (c) $-3y + 11$

4. Nested Brackets: Working from the Inside Out

When an expression contains brackets inside other brackets, different bracket symbols are used to make the layers clear:

- Round brackets: $( \dots )$

- Square brackets: $[ \dots ]$

- Curly brackets: $\{ \dots \}$ (for a third layer, not needed in this lesson)

Expressions with nested brackets should always be expanded from the inside out.

Procedure for Nested Brackets

1. Innermost first: Identify the inner round brackets $( \dots )$ and expand them using the distributive law.

2. Simplify inside: Combine like terms inside the outer square brackets $[ \dots ]$ before doing anything else.

3. Outermost bracket: Expand the outer square brackets by multiplying each term inside by the number outside (with its sign).

4. Final combination: Combine any remaining like terms outside the brackets.

Worked Example 5: Simplifying Nested Brackets

Expand and simplify each of the following:
(a) $2[p - 2(p + q)]$
(b) $5x - 3[2x - (x - 4)]$

Solution:

(a) First expand the inner bracket $-2(p + q)$ inside the square brackets:

$2[p - 2(p + q)] = 2[p - 2p - 2q]$

Next, combine like terms inside the square brackets: $p - 2p = -p$ :

$= 2[-p - 2q]$

Finally, multiply each term inside by $2$ :

$= 2(-p) + 2(-2q) = -2p - 4q$

(b) First expand the inner bracket $-(x - 4) = -x + 4$ inside the square brackets:

$5x - 3[2x - (x - 4)] = 5x - 3[2x - x + 4]$

Simplify inside the square brackets: $2x - x = x$ :

$= 5x - 3[x + 4]$

Multiply each term inside by $-3$ :

$= 5x - 3x - 12$

Combine like terms: $5x - 3x = 2x$ :

$= 2x - 12$

(a) $-2p - 4q$ (b) $2x - 12$

Common Mistakes to Avoid

Multiplying only the first term: Writing $3(2x + 5) = 6x + 5$ instead of multiplying $3 \times 5 = 15$ to get $6x + 15$ . Every term inside must be multiplied.
Sign errors with negative multipliers: Writing $-2(x - 4) = -2x - 8$ . Remember that $(-2) \times (-4) = +8$ , so the correct expansion is $-2x + 8$ .
Subtracting before multiplying: In $10 - 3(x + 2)$ , calculating $10 - 3 = 7$ first to get $7(x + 2)$ . Brackets and multiplication come before subtraction: $10 - 3x - 6 = -3x + 4$ .
Losing the minus sign on brackets: Writing $-(a - b) = -a - b$ . The minus sign distributes to both terms: $-1(a - b) = -a + b$ .
Omitting brackets around multi-term expressions before units: Writing $10x + 22\text{ m}$ instead of $(10x + 22)\text{ m}$ .

Practice Questions

Work out each problem on paper before checking your solution. Questions 9 to 12 apply expanding brackets to real-world geometric and practical settings.

Question 1: Single Brackets with Positive Multipliers

Expand each expression completely:
(a) $3(4x + 5)$
(b) $6(2y - 3)$
(c) $5(2a + 3b - 4c)$

ShowHide solution

(a) $3 \times 4x + 3 \times 5 = 12x + 15$ .

(b) $6 \times 2y - 6 \times 3 = 12y - 18$ .

(c) $5 \times 2a + 5 \times 3b - 5 \times 4c = 10a + 15b - 20c$ .

(a) $12x + 15$ (b) $12y - 18$ (c) $10a + 15b - 20c$

Question 2: Single Brackets with Negative Multipliers

Expand each expression completely:
(a) $-2(4x - 3)$
(b) $-5(2a - 3b + 1)$
(c) $-3(-2m + 5n - 4)$

ShowHide solution

(a) $(-2) \times 4x + (-2) \times (-3) = -8x + 6$ .

(b) $(-5) \times 2a + (-5) \times (-3b) + (-5) \times 1 = -10a + 15b - 5$ .

(c) $(-3) \times (-2m) + (-3) \times 5n + (-3) \times (-4) = 6m - 15n + 12$ .

(a) $-8x + 6$ (b) $-10a + 15b - 5$ (c) $6m - 15n + 12$

Question 3: Expanding Brackets with Leading Minus Signs

Expand each expression completely:
(a) $-(3x + 7)$
(b) $-(5a - 2b)$
(c) $-(p - 4q + 6)$

ShowHide solution

(a) $-1 \times (3x + 7) = -3x - 7$ .

(b) $-1 \times (5a - 2b) = -5a + 2b$ .

(c) $-1 \times (p - 4q + 6) = -p + 4q - 6$ .

(a) $-3x - 7$ (b) $-5a + 2b$ (c) $-p + 4q - 6$

Question 4: Expanding Multiple Brackets with Addition

Expand and simplify each expression completely:
(a) $2(3x + 1) + 4(x + 2)$
(b) $3(2a - 3) + 5(a + 4)$
(c) $4(2p - q) + 3(p + 2q)$

ShowHide solution

(a) $6x + 2 + 4x + 8 = (6x + 4x) + (2 + 8) = 10x + 10$ .

(b) $6a - 9 + 5a + 20 = (6a + 5a) + (-9 + 20) = 11a + 11$ .

(c) $8p - 4q + 3p + 6q = (8p + 3p) + (-4q + 6q) = 11p + 2q$ .

(a) $10x + 10$ (b) $11a + 11$ (c) $11p + 2q$

Question 5: Expanding Multiple Brackets with Subtraction

Expand and simplify each expression completely:
(a) $4(2x + 3) - 3(x - 2)$
(b) $5(3a - 1) - 2(4a + 5)$
(c) $6(u - 2v) - 2(3u - 5v)$

ShowHide solution

(a) $8x + 12 - 3x + 6 = (8x - 3x) + (12 + 6) = 5x + 18$ .

(b) $15a - 5 - 8a - 10 = (15a - 8a) + (-5 - 10) = 7a - 15$ .

(c) $6u - 12v - 6u + 10v = (6u - 6u) + (-12v + 10v) = -2v$ .

(a) $5x + 18$ (b) $7a - 15$ (c) $-2v$

Question 6: Order of Operations with Terms Outside Brackets

Expand and simplify each expression:
(a) $9 - 2(3x - 4)$
(b) $7a - 3(2a - 5) + 4$
(c) $12 - (4m - 3) + 2m$

ShowHide solution

(a) Multiply by $-2$ first: $9 - 6x + 8 = -6x + 17$ .

(b) Multiply by $-3$ first: $7a - 6a + 15 + 4 = a + 19$ .

(c) Distribute minus sign first: $12 - 4m + 3 + 2m = (-4m + 2m) + (12 + 3) = -2m + 15$ .

(a) $-6x + 17$ (b) $a + 19$ (c) $-2m + 15$

Question 7: Expanding Three Sets of Brackets

Expand and simplify completely:
(a) $2(x + 3) + 3(2x - 1) - 4(x + 2)$
(b) $3(2a - b) - 2(a - 3b) + 4(a + b)$

ShowHide solution

(a) Expand all three brackets:
$2x + 6 + 6x - 3 - 4x - 8$
Group like terms:
$(2x + 6x - 4x) + (6 - 3 - 8) = 4x - 5$

(b) Expand all three brackets:
$6a - 3b - 2a + 6b + 4a + 4b$
Group like terms:
$(6a - 2a + 4a) + (-3b + 6b + 4b) = 8a + 7b$

(a) $4x - 5$ (b) $8a + 7b$

Question 8: Nested Brackets from the Inside Out

Simplify each expression by expanding from the inside out:
(a) $3[x + 2(x - 4)]$
(b) $4[2a - (a - 3)]$
(c) $5x - 2[3x - 2(x - 1)]$

ShowHide solution

(a) Inner bracket first: $3[x + 2x - 8] = 3[3x - 8] = 9x - 24$ .

(b) Inner bracket first: $4[2a - a + 3] = 4[a + 3] = 4a + 12$ .

(c) Inner bracket first: $5x - 2[3x - 2x + 2] = 5x - 2[x + 2] = 5x - 2x - 4 = 3x - 4$ .

(a) $9x - 24$ (b) $4a + 12$ (c) $3x - 4$

Question 9: Poster Dimensions in a Geometry Setting

A rectangular poster has a length of $4(x + 3)\text{ cm}$ and a width of $(3x + 5)\text{ cm}$ .
(a) Write down an expression for the perimeter of the poster in terms of $x$ , and simplify it completely.
(b) If $x = 5$ , find the length, width, and perimeter of the poster in centimetres.

ShowHide solution

(a) The perimeter $P$ is given by $P = 2(\text{length} + \text{width})$ :
$\begin{aligned} P &= 2[4(x + 3) + (3x + 5)] \\ &= 2[4x + 12 + 3x + 5] \\ &= 2[7x + 17] \\ &= (14x + 34)\text{ cm} \end{aligned}$
Enclose the multi-term expression in brackets before the unit: $(14x + 34)\text{ cm}$ .

(b) Substitute $x = 5$ :
$\text{Length} = 4(5 + 3) = 4(8) = 32\text{ cm}$
$\text{Width} = 3(5) + 5 = 15 + 5 = 20\text{ cm}$
$\text{Perimeter} = 14(5) + 34 = 70 + 34 = 104\text{ cm}$
Check: $2(32 + 20) = 2(52) = 104\text{ cm}$ .

(a) $(14x + 34)\text{ cm}$ (b) Length $= 32\text{ cm}$ , Width $= 20\text{ cm}$ , Perimeter $= 104\text{ cm}$

Question 10: Cinema Ticket Promotion in a Setting

At a cinema, normal tickets cost $12 for an adult and $8 for a student.
During a promotion, each ticket receives a $2 discount.
A group buys $x$ adult tickets and $(2x + 3)$ student tickets.
(a) Write down the discounted price of one adult ticket and one student ticket.
(b) Write an expression in terms of $x$ for the total amount paid by the group, expanding and simplifying completely.
(c) If $x = 3$ , calculate the total amount paid by the group.

ShowHide solution

(a) Discounted adult ticket: $12 - 2 = 10$ , so $10.
Discounted student ticket: $8 - 2 = 6$ , so $6.

(b) Total cost $= 10x + 6(2x + 3)$ :
$\text{Total cost} = 10x + 12x + 18 = (22x + 18)\text{ dollars}$

(c) Substitute $x = 3$ into $22x + 18$ :
$\text{Total cost} = 22(3) + 18 = 66 + 18 = 84$
The group paid a total of $84.
Check: 3 adults at $10 each is $30. $2(3) + 3 = 9$ students at $6 each is $54. Total: $30 + 54 = 84$ , so $84.

(a) Adult: $10, Student: $6 (b) $(22x + 18)$ dollars (c) $84

Question 11: Two Garden Plots in a Geometry Setting

A landscape gardener designs two rectangular flower plots:

Plot 1 has perimeter $2[3(x + 2) + (x + 1)]\text{ m}$ .
Plot 2 has perimeter $2[2(x + 4) + (2x - 3)]\text{ m}$ .
(a) Expand and simplify the expression for the perimeter of Plot 1.
(b) Expand and simplify the expression for the perimeter of Plot 2.
(c) Find an expression for the total length of fencing needed to go round both plots, and evaluate it when $x = 3$ .

ShowHide solution

(a) For Plot 1:
$\begin{aligned} P_1 &= 2[3x + 6 + x + 1] \\ &= 2[4x + 7] \\ &= (8x + 14)\text{ m} \end{aligned}$

(b) For Plot 2:
$\begin{aligned} P_2 &= 2[2x + 8 + 2x - 3] \\ &= 2[4x + 5] \\ &= (8x + 10)\text{ m} \end{aligned}$

(c) Total length of fencing $=$ perimeter of Plot 1 $+$ perimeter of Plot 2:
$P_{\text{total}} = (8x + 14) + (8x + 10) = (16x + 24)\text{ m}$
Substitute $x = 3$ :
$16(3) + 24 = 48 + 24 = 72\text{ m}$
Check: $P_1 = 8(3) + 14 = 38\text{ m}$ ; $P_2 = 8(3) + 10 = 34\text{ m}$ ; $38 + 34 = 72\text{ m}$ .

(a) $(8x + 14)\text{ m}$ (b) $(8x + 10)\text{ m}$ (c) $(16x + 24)\text{ m}$ ; $72\text{ m}$

Question 12: Swimming Training in a Setting

During swimming training across three days, Bala swam lengths of a $25\text{ m}$ pool:

On Monday, he swam $3(2t + 5)$ lengths.
On Wednesday, he swam $4(3t - 2)$ lengths.
On Friday, he swam $2[5t - (t - 4)]$ lengths.
(a) Expand and simplify the expression for the lengths swum on Friday.
(b) Write down an expression in terms of $t$ for the total number of lengths Bala swam across all three days, in its simplest form.
(c) If $t = 3$ , calculate the total number of lengths swum.

ShowHide solution

(a) Expand the inner bracket first:
$2[5t - t + 4] = 2[4t + 4] = (8t + 8)\text{ lengths}$

(b) Sum the lengths for all three days:
$\text{Monday} = 3(2t + 5) = 6t + 15$
$\text{Wednesday} = 4(3t - 2) = 12t - 8$
$\text{Friday} = 8t + 8$
Add them together:
$\begin{aligned} \text{Total lengths} &= (6t + 15) + (12t - 8) + (8t + 8) \\ &= (6t + 12t + 8t) + (15 - 8 + 8) \\ &= (26t + 15)\text{ lengths} \end{aligned}$

(c) Substitute $t = 3$ into the simplified total:
$\text{Total} = 26(3) + 15 = 78 + 15 = 93\text{ lengths}$
Check each day: Monday $= 6(3) + 15 = 33$ ; Wednesday $= 12(3) - 8 = 28$ ; Friday $= 8(3) + 8 = 32$ .
$33 + 28 + 32 = 93\text{ lengths}$ , which is $93 \times 25 = 2325\text{ m}$ in all.

(a) $(8t + 8)$ lengths (b) $(26t + 15)$ lengths (c) $93$ lengths

Key Points to Remember

The Distributive Law: $a(b + c) = ab + ac$ and $a(b - c) = ab - ac$ . Multiply the outside number by every term inside the bracket.
Negative Multipliers Reverse Signs: When multiplying by a negative number, every sign inside the bracket flips: $-a(b - c) = -ab + ac$ .
An Isolated Minus Sign Means $-1$ : $-(x - y) = -1(x - y) = -x + y$ .
Order of Operations: multiply out a bracket before adding or subtracting the terms outside it; inside a bracket, simplify first.
Nested Brackets: Expand inner round brackets $( \dots )$ first, collect like terms inside the square brackets $[ \dots ]$ , and then expand outward.
The Unit Rule: When an algebraic expression has more than one term and carries a unit, always enclose the expression in brackets, such as $(14x + 34)\text{ cm}$ or $(22x + 18)$ dollars.