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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.

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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

  1. First: 2x × x = 2x²
  2. Outer: 2x × (−4) = −8x
  3. Inner: 3 × x = 3x
  4. Last: 3 × (−4) = −12
  5. Collect: 2x² − 8x + 3x − 12 = 2x² − 5x − 12
Answer2x² − 5x − 12
Example 2

Expand and simplify (x + 5)²

Working

  1. Use (a + b)² = a² + 2ab + b² with a = x, b = 5
  2. = x² + 2(x)(5) + 5²
  3. = 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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