How to Expand Brackets in Algebra: Single and Double Brackets Explained

We often encounter brackets in algebra, from simple expressions to more complex calculations. As we progress, we need to know how to expand them correctly, particularly when working with two brackets or expressions that involve negative numbers.

To help you approach these problems with confidence, we’ll work through single and double brackets step by step, explain the grid and FOIL methods and show how to recognise common mistakes in your working.

What Does Expanding Brackets Mean in Algebra?

To expand brackets in algebra, we apply multiplication to the expressions inside them and rewrite the result without brackets. The distributive law allows us to do this by multiplying each part of a sum or difference separately. 

For example, let's calculate 4(6 + 3). We can first add the numbers inside the brackets:

4(6 + 3) = 4 × 9 = 36

Alternatively, we can use the distributive law by multiplying both 6 and 3 by 4 before adding the results:

4 × 6 + 4 × 3 = 24 + 12 = 36

Both calculations give us 36, but the second shows how the distributive law allows us to expand the brackets without first calculating their contents. We follow the same rule in algebra, even when letters represent numbers whose values we don't know. 

In England, students learn to expand single brackets and products of two brackets during Key Stage 3, covering Years 7 to 9.

Bracket expansion is also part of GCSE algebra, as outlined in the AQA maths specification, one of England's exam boards. Students can also use AQA's past papers to practise exam-style maths questions.

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How to Expand Single Brackets

To expand a single bracket, multiply every term inside it by the number or letter outside, applying the usual rules for multiplying positive and negative numbers. Single brackets can contain numbers, letters or a combination of both. 

Let's see how this works with a positive multiplier first, before moving on to negative numbers and expressions containing letters. 

1. Expand a Bracket with a Positive Number

Suppose we want to expand 5(2x + 7). The number 5 multiplies both 2x and 7, even though the two terms are joined by an addition sign.

We can work through the calculation in three steps.

Step 1: Multiply the first term by 5

We multiply 5 by 2x. Since 5 × 2 = 10, we get 5 × 2x = 10x.

Step 2: Multiply the second term by 5

Next, we multiply 5 by 7 to get 5 × 7 = 35.

Step 3: Write the expanded expression

We put the two results together with the original addition sign to get 5(2x + 7) = 10x + 35.

2. Expand a Bracket with a Negative Number

We follow the same steps when the number outside is negative, but we also need to account for the signs of the numerical factors. Two numbers with opposite signs have a negative product, while two negative numbers have a positive product. 

Here's how we expand −4(3y − 2).

Step 1: Multiply −4 by 3y

We multiply −4 by the coefficient 3 and keep the y, giving −4 × 3y = −12y.

Step 2: Multiply −4 by −2

Both numerical factors are negative, so their product is positive: −4 × (−2) = +8.

Step 3: Write the expanded expression

We put the two products together to get −4(3y − 2) = −12y + 8.

3. Expand a Bracket with a Letter Outside

We may also encounter an expression such as 2x(x + 5), where the multiplier contains a letter. The calculation follows the same pattern, but we also multiply the letters. For example, x × x = x², which we read as “x squared”.

Here, we’ll expand 2x(x + 5) step by step.

Step 1: Multiply 2x by x

We multiply the numerical parts and the letters to get 2x × x = 2x².

Step 2: Multiply 2x by 5

We multiply 2 by 5 and keep the x, which gives 2x × 5 = 10x.

Step 3: Write the expanded expression

We put the two products together to get 2x(x + 5) = 2x² + 10x.

We leave 2x² and 10x as separate terms because they contain different powers of x and cannot be combined as like terms.

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How to Expand Double Brackets

To expand double brackets, we multiply each term in the first bracket by each term in the second, then collect any like terms. 

For example, (x + 4)(x − 7) means we multiply the two bracketed expressions together, which gives us four products to calculate.

We can organise the calculation using either the grid method or the FOIL method. Both give the same expanded expression, but they help us keep track of the multiplication in different ways.

1. Expand Double Brackets Using the Grid Method

To use the grid method, we arrange the terms from both brackets in a table, with one bracket's terms across the top and the other's down the left side. Each cell shows the product of its row and column headings. 

Let's expand (x + 4)(x − 7) using a 2 × 2 grid.

Step 1: Set Up the Grid

First, write x and 4 across the top, then x and −7 down the left side.

We now have four empty cells, one for each multiplication.

Step 2: Multiply the Terms to Fill Each Cell

For each cell, multiply the term at the top of its column by the term at the beginning of its row.

• Top-left: x × x = x²

• Top-right: x × 4 = 4x

• Bottom-left: −7 × x = −7x

• Bottom-right: −7 × 4 = −28

Our completed grid looks like this:

Step 3: Write the Expanded Expression

We write the four products as one expression, with plus signs for positive terms and minus signs for negative terms. We get x² + 4x − 7x − 28. 

Step 4: Collect Like Terms

We can combine 4x and −7x because both contain x to the first power. So, we get 4x − 7x = −3x. However, x² represents x × x, so it cannot be combined with terms containing only x. 

Our final answer is (x + 4)(x − 7) = x² − 3x − 28.

2. Expand Double Brackets Using the FOIL Method

FOIL stands for First, Outer, Inner and Last, which tells us the order in which to multiply the terms when expanding two brackets that each contain two terms. 

To see what each letter means, let's return to (x + 4)(x − 7) and expand it using FOIL.

Step 1: Multiply the First Terms

Multiply the first term in each bracket: x × x = x².

Step 2: Multiply the Outer Terms

Multiply the first term in the first bracket by the last term in the second: x × (−7) = −7x.

Step 3: Multiply the Inner Terms

Multiply the second term in the first bracket by the first term in the second: 4 × x = 4x.

Step 4: Multiply the Last Terms

Multiply the last term in each bracket: 4 × (−7) = −28.

We now have x² − 7x + 4x − 28. We can combine the like terms −7x and 4x to get x² − 3x − 28, the same answer we found using the grid method.

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3 Common Mistakes When Expanding Brackets

The three common mistakes when expanding brackets are missing a term, losing track of negative signs and squaring a bracket incorrectly. Let’s look at how to recognise each one in your working. 

1. Missing a Term When Multiplying

When we expand two brackets, we can overlook one of the four products, particularly if we move straight to simplifying the expression. 

For example, (x + 2)(x + 3) might be incorrectly expanded as x²+ 3x + 6. Here, we have missed 2 × x = 2x. 

We need to include 2x, so the correct expansion is x²+ 3x + 2x + 6 = x²+ 5x + 6. Before collecting like terms, check that you've calculated all four products. 

2. Losing Track of Negative Signs

Students may make sign errors when multiplying negative numbers, particularly when they forget that two negative factors produce a positive result. For example, −3(x − 4) expands to −3x + 12, not −3x − 12, because −3 × (−4) = +12.

You can check each product's sign by looking at whether the two factors have the same sign or opposite signs. 

3. Squaring Brackets Incorrectly

Students may assume that squaring a bracket means squaring each term separately. However, (x + 5)² means (x + 5)(x + 5), so we must multiply every term in the first bracket by every term in the second.

We get (x + 5)² = x²+ 5x + 5x + 25 = x²+ 10x + 25, not x² + 25. The two middle products, 5x and 5x, combine to give the 10x term that would otherwise be missing. 

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How Mathnasium Helps Students Understand Algebra and Beyond 

Mathnasium is a maths-only learning centre dedicated to helping students of all skill levels excel in maths.

When children learn to expand brackets, they apply multiplication across algebraic expressions. Our specially trained instructors help them build on what they already know, from working with numbers and negative signs to simplifying expressions and solving equations. 

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