Expanding double brackets Revision Notes for AQA GCSE Maths

Expanding double brackets

Expanding double brackets means multiplying two brackets together to create a single algebraic expression. This is a fundamental skill in algebra that you'll need to master for your GCSE exam. There are three main methods you can use, and it's helpful to know all three so you can choose the one that works best for each situation.

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Mastering double bracket expansion is essential for GCSE success. This skill appears frequently in exams and forms the foundation for more advanced algebraic techniques like factorising quadratics and solving equations.

Method 1: One-at-a-time method

The one-at-a-time method involves expanding one bracket first, then dealing with the other. This approach can be easier to understand when you're starting out with double bracket expansion.

How it works:

Take the first bracket and multiply each term in it by the entire second bracket
Expand the resulting expression as normal
Collect like terms to simplify your final answer

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Worked Example: Using the One-at-a-Time Method

Expand $(n + 3)(n + 8)$

Step 1: Multiply $n$ by $(n + 8)$ $n(n + 8) = n^2 + 8n$

Step 2: Multiply $3$ by $(n + 8)$ $3(n + 8) = 3n + 24$

Step 3: Add these together $n^2 + 8n + 3n + 24$

Step 4: Collect like terms $n^2 + 11n + 24$

Remember to always collect like terms at the end to get your expression in its simplest form.

Method 2: FOIL method

The FOIL method uses a systematic approach where you multiply terms in a specific order. FOIL stands for First, Outer, Inner, Last.

How it works:

First terms: Multiply the first term in each bracket
Outer terms: Multiply the outer terms (first term of first bracket × second term of second bracket)
Inner terms: Multiply the inner terms (second term of first bracket × first term of second bracket)
Last terms: Multiply the last term in each bracket

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Worked Example: Using the FOIL Method

Expand $(2a + b)(a - b)$

First: $2a \times a = 2a^2$
Outer: $2a \times (-b) = -2ab$
Inner: $b \times a = ab$
Last: $b \times (-b) = -b^2$

Combine: $2a^2 - 2ab + ab - b^2 = 2a^2 - ab - b^2$

Some students find it helpful to think of the FOIL pattern as making a "smiley face" when drawing the connections between terms.

Method 3: Grid method

The grid method uses a table to organise your multiplication systematically. This visual approach can be particularly helpful for avoiding mistakes.

How it works:

Draw a grid with the terms from the first bracket along the top
Write the terms from the second bracket down the side
Multiply each pair of terms and write the result in the corresponding grid square
Add all the results together and collect like terms

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Worked Example: Using the Grid Method

Expand $(x + 7)(x - 5)$

×	$x$	$-5$
$x$	$x^2$	$-5x$
$7$	$7x$	$-35$

Result: $x^2 - 5x + 7x - 35 = x^2 + 2x - 35$

When using the grid method, be careful with negative signs. Make sure you include them in your grid and apply them correctly when multiplying.

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Critical Success Tips

Always collect like terms at the end to simplify your answer completely
Watch out for negative signs - they can easily cause mistakes if you're not careful
Check your work by expanding a different way or substituting simple numbers
Practice all three methods so you can choose the one that feels most comfortable for different types of problems

When squaring a bracket like $(3p - 4)^2$ , remember that this means $(3p - 4)(3p - 4)$ , so you're multiplying the bracket by itself.

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Key Points to Remember:

Expanding double brackets means multiplying two brackets together to get a single expression
Three methods available: one-at-a-time, FOIL, and grid method - use whichever feels most comfortable
FOIL stands for First, Outer, Inner, Last - a systematic way to ensure you multiply all terms
Always collect like terms at the end to get your answer in its simplest form
Be extra careful with negative signs as they're a common source of errors in expansions

Expanding double brackets (AQA GCSE Maths): Revision Notes