What is important to know about Expanding double brackets?

FOIL: First × First, Outer × Outer, Inner × Inner, Last × Last — then collect like terms.

What is important to know about Expanding double brackets?

(a + b)² = a² + 2ab + b² — a common pattern to recognise.

What is important to know about Expanding double brackets?

(a + b)(a − b) = a² − b² — the difference of two squares.

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Algebra · Foundation & Higher

Expanding double brackets

Expanding double brackets involves multiplying every term in the first bracket by every term in the second bracket. The FOIL method (First, Outer, Inner, Last) is a helpful strategy.

Revising expanding double brackets the night before an exam? These are condensed revision notes — the facts, formulas and mistakes examiners actually mark, with no long explanations. Prefer a full walkthrough with step-by-step reasoning instead? Search “Expanding double brackets GCSE” on the site for the complete guide.

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Key facts to remember

1FOIL: First × First, Outer × Outer, Inner × Inner, Last × Last — then collect like terms.
2(a + b)² = a² + 2ab + b² — a common pattern to recognise.
3(a + b)(a − b) = a² − b² — the difference of two squares.
4Always collect the middle terms after expanding.
5The result of expanding two linear brackets is always a quadratic expression.

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Formulas

Perfect square

(a + b)² = a² + 2ab + b²

Difference of two squares

(a + b)(a − b) = a² − b²

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Worked examples

Example 1

Expand and simplify (2x + 3)(x − 4)

Working

First: 2x × x = 2x²
Outer: 2x × (−4) = −8x
Inner: 3 × x = 3x
Last: 3 × (−4) = −12
Collect: 2x² − 8x + 3x − 12 = 2x² − 5x − 12

Answer2x² − 5x − 12

Example 2

Expand and simplify (x + 5)²

Working

Use (a + b)² = a² + 2ab + b² with a = x, b = 5
= x² + 2(x)(5) + 5²
= x² + 10x + 25

Answerx² + 10x + 25

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Common mistakes

✗(x + 5)² ≠ x² + 25 — students forget the middle term 2 × x × 5.

✗Sign errors when one or both terms are negative.

✗Not collecting the two middle terms — leaving four terms instead of three.

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Exam tips

✓Use the grid method if you find FOIL confusing: draw a 2×2 grid and fill each cell.

✓Always write out all four products before collecting — never try to do it in one step.

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