Expanding Brackets –

Expanding brackets is a core algebra skill tested at both Foundation and Higher tier in GCSE Maths. It involves multiplying out expressions to remove brackets, a process you will use when solving equations, simplifying expressions, proving algebraic identities, and working with quadratics. Foundation students need to expand single brackets and simple double brackets; Higher students also meet triple brackets and expressions involving surds. This page gives you systematic methods for every type, worked examples at both tiers, the common traps to watch for, and practice questions for self-testing. For a broader revision plan, check our how to revise GCSE Maths guide.

What Does Expanding Brackets Mean?

Expanding (or multiplying out) brackets means using the distributive law to remove the brackets from an expression. Each term inside the bracket is multiplied by the term outside (for single brackets) or by every term in the other bracket (for double brackets).

Key Formulas

The double-bracket expansion is sometimes remembered using the acronym FOIL: First, Outside, Inside, Last — referring to the four multiplications you need to do.

Step-by-Step Method

Expanding a Single Bracket

1. Multiply the term outside the bracket by the first term inside.
2. Multiply the term outside the bracket by the second term inside.
3. Continue for every term inside the bracket.
4. Simplify if possible.

Example: Expand 3(2x + 5).

• 3 × 2x = 6x
• 3 × 5 = 15
• Result: 6x + 15

Expanding Double Brackets

1. Multiply the first term in the first bracket by both terms in the second bracket.
2. Multiply the second term in the first bracket by both terms in the second bracket.
3. Write out all four terms.
4. Collect like terms to simplify.

Example: Expand (x + 3)(x + 7).

• x × x = x²
• x × 7 = 7x
• 3 × x = 3x
• 3 × 7 = 21
• Collect like terms: x² + 7x + 3x + 21 = x² + 10x + 21

Expanding and Simplifying Two Single Brackets

Sometimes you need to expand two separate single brackets and then collect like terms:

Example: Expand and simplify 4(x + 2) − 3(x − 5).

• 4(x + 2) = 4x + 8
• −3(x − 5) = −3x + 15 (note: −3 × −5 = +15)
• Collect: 4x − 3x + 8 + 15 = x + 23

Squaring a Bracket (Higher)

A very common error is writing (x + 3)² = x² + 9. This is wrong because you must expand it as (x + 3)(x + 3):

1. Use the double bracket method: x² + 3x + 3x + 9.
2. Simplify: x² + 6x + 9.

Alternatively, use the formula: (a + b)² = a² + 2ab + b².

Expanding Triple Brackets (Higher)

1. First expand two of the three brackets using the double bracket method.
2. Then expand the result by the third bracket using single bracket expansion on each term.
3. Collect all like terms.

Worked Example 1 — Foundation Level

Question: Expand and simplify 5(2x − 3) + 2(x + 4).

Working:

Step 1 — Expand the first bracket: 5 × 2x = 10x, 5 × (−3) = −15. Result: 10x − 15.

Step 2 — Expand the second bracket: 2 × x = 2x, 2 × 4 = 8. Result: 2x + 8.

Step 3 — Combine: 10x − 15 + 2x + 8.

Step 4 — Collect like terms: (10x + 2x) + (−15 + 8) = 12x − 7.

Answer: 12x − 7

Worked Example 2 — Higher Level

Question: Expand and simplify (2x − 3)(x + 5).

Working:

Step 1 — First × First: 2x × x = 2x².

Step 2 — First × Second: 2x × 5 = 10x.

Step 3 — Second × First: (−3) × x = −3x.

Step 4 — Second × Second: (−3) × 5 = −15.

Step 5 — Write all terms: 2x² + 10x − 3x − 15.

Step 6 — Collect like terms: 2x² + 7x − 15.

Answer: 2x² + 7x − 15

Common Mistakes

• Not multiplying every term inside the bracket. In 3(x + 4), students sometimes write 3x + 4 instead of 3x + 12. The 3 must multiply both x and 4.
• Sign errors with negative terms. When expanding −2(x − 5), remember: −2 × x = −2x and −2 × (−5) = +10. A negative times a negative is a positive.
• Writing (x + 3)² as x² + 9. You must expand as (x + 3)(x + 3) = x² + 6x + 9. The middle term (2ab) is always present. See simplifying expressions for more on combining like terms after expanding.
• Forgetting to collect like terms. After expanding double brackets, you will usually have two like terms in the middle that need combining. Leaving them uncombined when the question says "simplify" will cost marks.
• Mixing up signs in the FOIL method. Track each multiplication carefully. Writing out F, O, I, L labels beside your working can help.

Exam Tips

• Use a grid or table for double brackets if FOIL confuses you. Draw a 2×2 grid with one bracket across the top and the other down the side. Multiply to fill in the four cells, then add them up. This is a reliable, visual method.
• For "show that" or "prove" questions, you must show each expansion step. Jumping to the final answer without showing the multiplication earns zero marks even if the answer is correct.
• Practise squaring brackets until (a + b)² = a² + 2ab + b² is automatic. This pattern appears in completing the square, the quadratic formula, and circle equations. Our formulas guide lists the identities to learn.
• Check your expansion by substituting a value. For example, expand (x + 2)(x + 3) to get x² + 5x + 6. Substitute x = 1: (3)(4) = 12, and 1 + 5 + 6 = 12. They match, so the expansion is correct.

Practice Questions

Q1 (Foundation): Expand 4(3x − 2).

Q2 (Foundation): Expand and simplify (x + 6)(x − 2).

Q3 (Higher): Expand and simplify (3x + 1)².

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Related Topics

• Simplifying Expressions
• Indices and Index Laws
• Surds
• Order of Operations (BIDMAS)

Summary

• Expanding brackets means multiplying each term inside by the term(s) outside.
• For single brackets, multiply the outside term by every term inside: a(b + c) = ab + ac.
• For double brackets, multiply every term in the first bracket by every term in the second (FOIL or grid method).
• (a + b)² = a² + 2ab + b² — never forget the middle term.
• (a + b)(a − b) = a² − b² — the difference of two squares has no middle term.
• Always collect like terms after expanding.
• Check your answer by substituting a simple value for the variable.