Expanding double brackets Revision Notes for Edexcel GCSE Maths

Expanding double brackets

When you have two brackets multiplied together, you need to expand them to create a single expression. This means multiplying every term in the first bracket by every term in the second bracket.

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Expanding brackets is a fundamental algebraic skill that forms the foundation for solving quadratic equations, factoring expressions, and many other advanced mathematical concepts.

There are three main methods you can use to expand double brackets. Choose whichever method works best for you.

Method 1: One-at-a-time

This method involves expanding one bracket first, then multiplying by the other bracket. It's often the most intuitive approach for beginners.

How it works:

Take the first bracket and multiply it by each term in the second bracket
Expand each part separately
Collect like terms at the end

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

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

Step 1: Multiply $(n + 3)$ by $n$ : $n(n + 3) = n^2 + 3n$

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

Step 3: Combine the results: $n^2 + 3n + 8n + 24$

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

Method 2: FOIL

FOIL stands for First, Outer, Inner, Last. This tells you which terms to multiply together and provides a systematic way to ensure you don't miss any combinations.

How it works:

First: Multiply the first terms in each bracket
Outer: Multiply the outer terms
Inner: Multiply the inner terms
Last: Multiply the last terms
Add all four results together and collect like terms

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Worked Example: Using the FOIL Method

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

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

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

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Some students remember the FOIL pattern as a "smiley face" when drawing the connections between terms. Try drawing curved lines to connect the terms you're multiplying - it can help you visualise the process!

Method 3: Grid method

The grid method uses a table to organise your multiplication systematically. This visual approach helps ensure you multiply every term correctly.

How it works:

Draw a grid with the terms from each bracket
Multiply each term in the first bracket by each term in the second bracket
Fill in all the boxes in your grid
Add all the results together and collect like terms

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Worked Example: Using the Grid Method

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

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

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

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Remember to be careful with negative signs - they belong to the number and must be written in your grid. A common mistake is forgetting to include the negative sign when copying terms into the grid.

Squaring brackets

When you square a bracket, you multiply it by itself. This is a special case of expanding double brackets where both brackets are identical.

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Worked Example: Squaring Brackets

$(3p - 4)^2$ means $(3p - 4)(3p - 4)$

Using the grid method:

×	$3p$	$-4$
$3p$	$9p^2$	$-12p$
$-4$	$-12p$	$16$

Result: $9p^2 - 12p - 12p + 16 = 9p^2 - 24p + 16$

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Remember that $3p \times 3p = 9p^2$ and be careful with negative signs: $(-4) \times (-4) = +16$ . When squaring negative terms, the result is always positive.

Collecting like terms

Always finish by collecting like terms to simplify your answer. Like terms have the same letter and power - they can be combined by adding or subtracting their coefficients.

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Example: Collecting Like Terms

$y^2 + 9y + 2y + 18$

The like terms are $9y$ and $2y$ (both have $y$ to the power of 1)

Final answer: $y^2 + 11y + 18$

Exam tips

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Essential Exam Strategies:

Show your working clearly - you can gain marks even if your final answer is wrong
Double-check your signs, especially with negative numbers
Always collect like terms as your final step
If the question asks you to "expand and simplify", make sure you do both parts
Take your time with the arithmetic - most errors occur in the calculation steps, not the method

Remember!

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Key Points to Remember:

You can expand double brackets using one-at-a-time, FOIL, or the grid method
FOIL stands for First, Outer, Inner, Last terms
Always collect like terms to simplify your final answer
When squaring brackets, you multiply the bracket by itself
Be extra careful with negative signs throughout your working
Choose the method that works best for you - they all give the same answer when done correctly!

Expanding double brackets (Edexcel GCSE Maths): Revision Notes