Lesson 8 Unit 2 · Patterns & Algebra
Expanding Single Brackets
$2(x+3)$ becomes $2x+6$. Learn the distributive law — the most important rule in algebra — and never mess up bracket expansion again.
Today's hook: A rectangle has width 2 and length $(x+3)$. Its area is $2(x+3)$. But you can also split it into two smaller rectangles. What are their areas?
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Expand $3(x + 4)$. Draw a diagram showing two rectangles that represent this expression.
1
The Big Idea
Expanding means multiplying the term outside the bracket by every term inside . This is the distributive law : $a(b+c) = ab + ac$.
$2(x+3)$: multiply 2 by $x$ = $2x$, then 2 by $+3$ = $+6$. Result: $2x + 6$ . The outside term visits every inside term.
$2$ width
$x$ $2 \times x = 2x$
$3$ $2 \times 3 = 6$
$x+3$ length
$2(x+3) = 2x + 6$
Area = width × length
$a(b+c) = ab + ac$
Multiply EVERY term
$3(x+2) = 3x + 6$. The 3 multiplies both $x$ AND $2$.
Watch the signs
$-2(x-3) = -2x + 6$. Negative times negative = positive.
Area model helps
Think of a rectangle split into parts. Total area = sum of parts.
2
What You'll Master
objectives
Know The distributive law How to expand positive and negative terms The area model Understand Why every inside term gets multiplied How signs affect expansion That expansion and factorising are opposites Can Do Expand any single bracket expression Handle negative multipliers Check answers by substitution
3
Words You Need
vocabulary
Expand Remove brackets by multiplying the outside term by every term inside.
Distributive Law $a(b+c) = ab + ac$. The outside term is shared with every inside term.
Factorise The opposite of expanding. Putting a common factor outside brackets.
Area Model A rectangle diagram showing expansion as the sum of smaller areas.
Multiplier The term outside the bracket. e.g. In $3(x+2)$, the multiplier is 3.
Terms Inside The expressions within the brackets. e.g. In $2(x+5)$, the terms are $x$ and $+5$.
Right: $3(x+2) = 3x + 6$. The 3 multiplies BOTH terms.
Wrong: $-2(x-3) = -2x - 6$
Right: $-2(x-3) = -2x + 6$. Neg × neg = pos!
5
The Distributive Law
Multiply the outside term by every term inside. This is the most important rule in algebra.
$2(x+5)$
$= 2x + 10$
$2 \times x = 2x$, $2 \times 5 = 10$
$3(2x-1)$
$= 6x - 3$
$3 \times 2x = 6x$, $3 \times (-1) = -3$
$-4(x+3)$
$= -4x - 12$
$-4 \times x = -4x$, $-4 \times 3 = -12$
$-2(3x-5) = -6x + 10$
$-2 \times 3x = -6x$ $-2 \times (-5) = +10$
Neg × neg = pos. This is the most common mistake!
Draw arrows from the outside term to each inside term. Follow each arrow and multiply. You can't miss a term!
$3$
$(2x - 5)$
$3 \times 2x = 6x$
$3 \times (-5) = -15$
$= 6x - 15$
Try It: Expand $-2(4x + 3)$ using arrows.Ans: $-8x - 6$
7
Checking Your Answer
Pick a value for $x$ and check both the original and expanded forms give the same answer.
$3(x+2)$ when $x=4$
$3(4+2) = 3(6) = 18$
Original
$3x + 6$ when $x=4$
$3(4) + 6 = 12 + 6 = 18$
Expanded
Both give 18 ✓ Expansion is correct!
8
Quick Reference Table
Expression Expansion
$2(x+3)$ $2x + 6$ $3(2x-1)$ $6x - 3$ $-4(x+2)$ $-4x - 8$ $-2(3x-5)$ $-6x + 10$ $x(x+3)$ $x^2 + 3x$
Watch Me Solve It · Worked example
Watch Me Solve It · Negative expansion
Q
PROBLEM
Expand $-3(2x - 4)$.
1
Identify the multiplier and inside terms
Multiplier: $-3$. Inside: $2x$ and $-4$
The bracket has two terms: $2x$ (positive) and $-4$ (negative).
2
Multiply $-3$ by $2x$
$-3 \times 2x = -6x$
Neg × pos = neg. $3 \times 2 = 6$. So $-6x$.
3
Multiply $-3$ by $-4$
$-3 \times (-4) = +12$
Neg × neg = pos! $3 \times 4 = 12$. This is the step everyone gets wrong.
4
Combine the results
$-6x + 12$
Check: let $x=1$. Original: $-3(2-4) = -3(-2) = 6$. Answer: $-6+12 = 6$ ✓
Show me the next step
Answer $-6x + 12$
$3(x+2) = 3x + 2$
Only multiplied the first term! The 2 needs multiplying too.
Fix: draw arrows. $3 \times x = 3x$ AND $3 \times 2 = 6$.
$-2(x-3) = -2x - 6$
Forgot that neg × neg = pos. $-2 \times (-3) = +6$.
Fix: track signs carefully. Write $-2 \times (-3) = +6$ explicitly.
$x(x+4) = x + 4x = 5x$
Forgot that $x \times x = x^2$, not $x$.
Fix: $x \times x = x^2$. The answer is $x^2 + 4x$.
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Brain Trainer · 4 problems
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Brain Trainer · Mixed
4 problems
1 Expand $4(x + 3)$.
Show answer $4x + 12$$4x + 12$
2 Expand $2(3x - 5)$.
Show answer $6x - 10$$6x - 10$
3 Expand $-3(2x + 1)$.
Show answer $-6x - 3$. All signs flip!$-6x - 3$
4 Expand $-2(x - 4)$.
Show answer $-2x + 8$. Neg × neg = pos!$-2x + 8$
$3(x + 4) = $
A OPTION A $3x + 4$ B OPTION B $3x + 12$ C OPTION C $3x + 4$ D OPTION D $x + 12$
$2(3x - 1) = $
A OPTION A $5x - 1$ B OPTION B $6x - 2$ C OPTION C $6x - 2$ D OPTION D $6x - 1$
$-4(x + 2) = $
A OPTION A $-4x - 8$ B OPTION B $-4x + 8$ C OPTION C $4x - 8$ D OPTION D $-4x + 2$
$-2(3x - 5) = $
A OPTION A $-6x - 10$ B OPTION B $6x + 10$ C OPTION C $-6x - 5$ D OPTION D $-6x + 10$
$x(x + 3) = $
A OPTION A $x + 3$ B OPTION B $x^2 + 3$ C OPTION C $x^2 + 3x$ D OPTION D $2x + 3$
Show Your Working · 3 questions
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9 marks total
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Medium
2 MARKS
Q6. Expand $5(2x + 3)$.
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Medium
3 MARKS
Q7. Expand $-3(4x - 2)$ and check by substituting $x=1$.
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Medium
4 MARKS
Q8. A rectangle has width $x$ and length $(x+5)$. Write and simplify an expression for its area. Then find the area when $x=3$.
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Hard
3 MARKS
Stretch. Expand $3(2x+1) - 2(x-4)$.
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