Expanding brackets Revision Notes for AQA GCSE Maths

Expanding brackets

What is expanding brackets?

Expanding brackets means multiplying the term outside the brackets by every term inside the brackets. This process is also called multiplying out brackets.

When you expand brackets, you remove the brackets by distributing the outside term to each term within the brackets.

The golden rule

The concept of expanding brackets follows a fundamental principle that must be applied consistently in all cases.

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The Golden Rule for Expanding Brackets

You must multiply the expression outside the brackets by everything inside the brackets.

For example:

$4n \times n = 4n^2$
$4n \times 2 = 8n$

So: $4n(n + 2) = 4n^2 + 8n$

This rule applies to all bracket expansion problems. Make sure you multiply the outside term by each and every term inside the brackets.

Dealing with negative signs

Working with negative signs requires special attention as they can completely change the outcome of your calculations.

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Handling Negative Signs

You need to be extra careful when there are negative signs outside the brackets.

When expanding $-2(x - y)$ :

$-2 \times x = -2x$
$-2 \times (-y) = +2y$

Therefore: $-2(x - y) = -2x + 2y$

Remember the rules for multiplying with negative numbers:

When both terms are positive OR both terms are negative → the answer is positive
When one term is positive and one term is negative → the answer is negative

For example: $-2 \times (-4) = +8$ and $-10p \times 5q = -50pq$

Expanding and simplifying

Sometimes you need to expand and simplify. This means multiply out the brackets and then collect like terms.

The process involves two distinct stages: first expanding all brackets, then combining terms that are alike.

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Worked Example: Expanding and Simplifying

Expand and simplify: $6(m + 2) + 4(3 - m)$

Step 1: Expand both brackets

$6 \times m = 6m$ and $6 \times 2 = 12$
$4 \times 3 = 12$ and $4 \times (-m) = -4m$

Step 2: Write out the full expansion $6m + 12 + 12 - 4m$

Step 3: Collect like terms

Like terms with $m$ : $6m - 4m = 2m$
Number terms: $12 + 12 = 24$

Final answer: $2m + 24$

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Remember that any negative signs belong to the term on their right.

Worked examples

Here are additional examples that demonstrate the key techniques for expanding brackets:

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Example 1: Basic Expansion

Expand $5(2y - 3)$

Solution:

$5 \times 2y = 10y$
$5 \times (-3) = -15$

Answer: $10y - 15$

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Example 2: Expand and Simplify with Two Brackets

Expand and simplify $2(3x + 4) - 3(4x - 5)$

Solution:

First bracket: $2 \times 3x = 6x$ and $2 \times 4 = 8$
Second bracket: $-3 \times 4x = -12x$ and $-3 \times (-5) = +15$
Full expansion: $6x + 8 - 12x + 15$
Collect like terms: $6x - 12x = -6x$ and $8 + 15 = 23$

Answer: $-6x + 23$

Exam tips

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Key Strategies for Success:

Always multiply the outside term by every term inside the brackets
Be extra careful with negative signs - they can change the signs of all terms
When you see "expand and simplify", remember to collect like terms after expanding
Check your signs carefully when dealing with negative terms
Show all your working steps clearly
Double-check your final answer by substituting a simple value

Remember!

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Essential Points to Remember:

Expanding brackets means multiplying the outside term by everything inside the brackets
The golden rule is to multiply by every single term inside
Negative signs outside brackets change the signs of all terms inside
Expand and simplify means expand first, then collect like terms
Always double-check your signs when working with negative numbers

Expanding brackets (AQA GCSE Maths): Revision Notes