How to Expand Double Brackets: A Step-by-Step GCSE Maths Guide

Expanding double brackets is a fundamental algebra skill you'll need for your GCSE maths exams (AQA, Edexcel, OCR). It's the process of removing brackets from an expression like (x + 3)(x + 5) to write it as a single, simplified quadratic expression. Mastering this is essential for solving equations, sketching graphs, and tackling higher-level topics.

Let's break it down into simple, manageable steps.

What Does 'Expanding Double Brackets' Mean?

When you see two brackets next to each other, like (a + b)(c + d), it means you need to multiply everything in the first bracket by everything in the second bracket. The result will usually be a quadratic expression (an expression where the highest power of x is 2).

Method 1: The FOIL Method (A Reliable Classic)

FOIL is a handy acronym to help you remember the order of multiplication. It stands for First, Outer, Inner, Last.

Let's expand (x + 4)(x + 2).

First: Multiply the first terms in each bracket.

x × x = x²

Outer: Multiply the outer terms in the product.

x × 2 = 2x

Inner: Multiply the inner terms.

4 × x = 4x

Last: Multiply the last terms in each bracket.

4 × 2 = 8

Now, write them all out: x² + 2x + 4x + 8

Finally, collect like terms (the x terms): x² + 6x + 8

And there's your expanded expression!

Method 2: The Grid Method (A Visual Alternative)

If you prefer a more visual, structured approach, the grid method is excellent. Let's use the same example: (x + 4)(x + 2).

Draw a 2-by-2 grid. Write the terms from the first bracket across the top, and the terms from the second bracket down the side.

flowchart TD
    A["“Start: (x + 4)(x + 2)”"] --> B["“Draw a 2x2 grid”"]
    B --> C
    subgraph C["Grid"]
        direction LR
        D1["(blank)"] --> E1["x"] --> F1["+4"]
        D2["x"] --> E2["x²"] --> F2["4x"]
        D3["+2"] --> E3["2x"] --> F3["8"]
    end
    C --> G["“Add all results: x² + 4x + 2x + 8”"]
    G --> H["“Collect like terms: x² + 6x + 8”"]

Multiply each cell and then add all the results together: x² + 4x + 2x + 8 = x² + 6x + 8.

Both methods give you the same answer. Choose the one that feels most intuitive for you!

Handling Negative Terms

This is where many students stumble, but careful attention to signs is key. Let's expand (x - 5)(x + 3) using FOIL.

First: x × x = x²

Outer: x × (+3) = +3x

Inner: (-5) × x = -5x

Last: (-5) × (+3) = -15

We get: x² + 3x - 5x - 15

Collect like terms: x² - 2x - 15

Crucial Tip: The last term comes from -5 × +3 = -15. Always use the sign in front of the number when multiplying.

A Trickier Example: Two Negatives

Expand (x - 4)(x - 7).

First: x × x = x²

Outer: x × (-7) = -7x

Inner: (-4) × x = -4x

Last: (-4) × (-7) = +28 (A negative times a negative gives a positive!)

We get: x² - 7x - 4x + 28 = x² - 11x + 28

Squaring a Bracket: A Special Case

(x + 6)² is the same as (x + 6)(x + 6). Expand it as normal.

Common Mistake Alert: (x + 6)² is NOT x² + 36. You must expand the brackets fully.

(x + 6)² = (x + 6)(x + 6) = x² + 6x + 6x + 36 = x² + 12x + 36

Practice Makes Perfect: Your Turn

Try these examples. The answers are at the bottom of the post.

(x + 1)(x + 9)

(x + 5)(x - 2)

(x - 8)(x - 3)

(x + 10)(x - 10) (This is a 'difference of two squares' – spot the pattern!)

(2x + 1)(x + 4) (Hint: Just multiply each term as before: 2x × x = 2x²)

Why Is This Important for GCSE?

Expanding brackets is not an isolated skill. You will use it constantly:

Solving quadratic equations: Before you can solve (x+2)(x-1)=0, you might need to expand and rearrange from a different form.

Sketching quadratic graphs: The expanded form ax² + bx + c helps you find the y-intercept.

Algebraic proof: You'll often need to expand and simplify to prove a given relationship.

Summary and Exam Tips

Choose your weapon: Decide on FOIL or the grid method and stick with it for consistency.

Signs are crucial: Pay meticulous attention to positive and negative signs when multiplying.

Always collect like terms: Your final answer should be a neat, simplified expression.

Check your work: A quick mental check (e.g., if x=1, does your original bracket give the same value as your expanded answer?) can catch errors.

Look for patterns: Notice that (x+a)(x-a) always gives x² - a² (difference of two squares).

Mastering double bracket expansion builds a strong foundation for the rest of your GCSE algebra. Practise regularly, and it will soon become second nature.

Answers to Practice Questions:

x² + 10x + 9

x² + 3x - 10

x² - 11x + 24

x² - 100 (The +10x and -10x cancel out)

2x² + 9x + 4
Need more help? Search for past paper questions on 'expanding brackets' from your exam board (AQA, Edexcel, OCR) for the best practice.