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.
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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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