Expanding Brackets

Expanding brackets is a key algebra skill in GCSE Maths used to simplify algebraic expressions and solve equations. Expanding brackets means multiplying everything inside the bracket by the term outside. In this lesson, we will learn how to expand single and double brackets step by step using clear examples and practice questions.

What does expanding brackets mean?

Expanding brackets removes the bracket by multiplying each term inside it.

Examples:

3(x+4)3(x + 4)

2(a−5)2(a − 5)

x(x+3)x(x + 3)

We multiply the outside term by every term inside the bracket.

Example Question:

Expand:

3(x+4)3(x + 4)

Exam Tip

Always multiply every term inside the bracket — not just the first one.

Missing one term is the most common mistake in exams.

Look for:

  • Single brackets
  • Negative signs
  • Terms with variables

Method – Expanding Single Brackets

To expand a bracket:

  1. Multiply the outside term by the first term inside
  2. Multiply the outside term by the second term inside
  3. Write the results and simplify if possible

Worked Example

Expand:

3(x+4)3(x + 4)

Step 1: Multiply 33 by xx

3×x=3x3 × x = 3x

Step 2: Multiply 33 by 44

3×4=123 × 4 = 12

Step 3: Write the result

3x+123x + 12

Worked Example 2

Expand:

5(a−2)5(a − 2)

Step 1: Multiply 55 by aa

5a5a

Step 2: Multiply 55 by −2−2

5×−2=−105 × −2 = −10

Step 3: Write the result

5a−105a − 10

Worked Example 3 (With Variables)

Expand:

x(x+3)x(x + 3)

Step 1: Multiply xx by xx

x×x=x2x × x = x²

Step 2: Multiply xx by 33

3x3x

Step 3: Write the result

x2+3xx² + 3x

Worked Example 4 (Negative Outside)

Expand:

−2(x+5)−2(x + 5)

Step 1: Multiply −2−2 by xx

−2x−2x

Step 2: Multiply −2−2 by 55

−10−10

Step 3: Write the result

−2x−10−2x − 10

Expanding double brackets

Multiply every term in the first bracket by every term in the second bracket. For two terms in each bracket, this gives four products. Keep the signs with the terms, then collect like terms.

Worked Example 5 (Two positive terms)

Expand and simplify: (x + 3)(x + 2)

Step 1: Multiply x by both terms: x × x + x × 2 = x² + 2x.

Step 2: Multiply 3 by both terms: 3 × x + 3 × 2 = 3x + 6.

Step 3: Combine the four products: x² + 2x + 3x + 6.

Answer: x² + 5x + 6. Only the 2x and 3x terms combine.

Worked Example 6 (A coefficient and a negative term)

Expand and simplify: (2x − 1)(x + 4)

Step 1: Multiply 2x by both terms: 2x × x + 2x × 4 = 2x² + 8x.

Step 2: Multiply −1 by both terms: −1 × x + (−1) × 4 = −x − 4.

Step 3: Collect like terms: 2x² + 8x − x − 4.

Answer: 2x² + 7x − 4. Remember that the negative sign belongs to the −1.

Final answer:

The expanded expression contains no brackets and is fully simplified.

Check your solution (important habit)

Ensure:

• Every term inside the bracket was multiplied

• Signs are correct

• Like terms are simplified if needed

Quick check example:

2(x+3)→2x+62(x + 3) → 2x + 6 ✔️

2(x+3)→2x+32(x + 3) → 2x + 3 ❌

Think Like a Mathematician

Expanding brackets is based on the distributive law.

This means multiplication is “distributed” across addition or subtraction inside the bracket.

Understanding this helps when working with more complex algebra later.

Common mistakes to avoid

Forgetting to multiply the second term

• Getting negative signs wrong

• Not simplifying the result

• Mixing up multiplication and addition

Negative numbers cause most errors — slow down and check signs carefully.

Try one yourself

Expand:

4(x−3)4(x − 3)

Multiply 4 by each term inside the bracket.

4 × x = 4x and 4 × (−3) = −12.

Answer: 4x − 12.

Now try double brackets: expand and simplify (x − 2)(x + 5).

Multiply each term in the first bracket by both terms in the second.

(x − 2)(x + 5) = x² + 5x − 2x − 10.

Collect the x terms: 5x − 2x = 3x.

Answer: x² + 3x − 10.

Frequently Asked Questions

What does expanding brackets mean?

Expanding brackets means multiplying the terms inside the brackets by the term outside to remove the brackets.

What is the difference between single and double brackets?

Single brackets involve multiplying one term by several terms, while double brackets involve multiplying two expressions together.

Why does x×xx × x become x2x²?

Because multiplying a variable by itself gives the variable squared.

x×x=x2x × x = x²

Next Lesson

Continue → Factorising Expressions
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