Factorising Quadratics into Two Brackets · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Factorising Quadratics into Two Brackets

Mathematics · Algebra · ages 14-15
Name ______________________   Date ____________

  1. When c is positive and b is negative, the pair uses two negative numbers.

    Circle one:   True   False

  2. Which pair multiplies to -6 and adds to 1?

    • 2 and -3
    • 1 and -6
    • -2 and 3

  3. Amir factorised x² - 3x - 10 as (x-5)(x+2). He checks it by expanding the brackets. True or false: his factorisation is correct.

    Circle one:   True   False

  4. In x squared + 7x + 12, the pair must multiply to 12 and add to 7. Which pair is it?

    • 3 and 4
    • 2 and 6
    • 12 and 1

  5. Factorise x² - 9x + 14 into two brackets.

    • (x+2)(x+7)
    • (x-1)(x-14)
    • (x-2)(x+7)
    • (x-2)(x-7)

  6. Sam factorises x squared + 9x + 20 and writes (x + 2)(x + 10), because 2 times 10 = 20. What went wrong?

    • 2 and 10 multiply to 20 but add to 12, not 9
    • The brackets should both carry negative signs
    • He should have split the x squared term into two parts

  7. Factorise x squared - 10x + 21 as (x - a)(x - b) with a smaller than b. What is a?

    Answer: ______________

  8. x squared - 7x + 12 factorises to (x - 3)(x - 4).

    Circle one:   True   False

  9. Which of these quadratics cannot be factorised over the integers?

    • x squared + 5x + 6
    • x squared - 4x - 21
    • x squared + 3x - 18
    • x squared + x + 2

  10. Factorise x squared - 12x + 35 as (x - a)(x - b) with a smaller than b. What is b - a?

    Answer: ______________

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Factorising Quadratics into Two Brackets W1-mt_3QXgmWKfoA-s1

  1. True · A positive c needs a pair with the same sign, and a negative b means both must be negative.
  2. -2 and 3 · -2 and 3 multiply to -6 and add to 1, so a quadratic like x squared + x - 6 factorises to (x - 2)(x + 3).
  3. True · Expanding (x-5)(x+2) gives back x² - 3x - 10, so Amir is right.
  4. 3 and 4 · 3 and 4 multiply to 12 and add to 7, so the factorisation is (x + 3)(x + 4).
  5. (x-2)(x-7) · The constant is positive but b is negative, so both numbers are negative. -2 and -7 multiply to 14 and add to -9.
  6. 2 and 10 multiply to 20 but add to 12, not 9 · The pair must do both jobs. 2 and 10 hit the product but add to 12, while 4 and 5 adds to 9.
  7. 3 · The pair is -3 and -7, so the brackets are (x - 3)(x - 7) and a is 3.
  8. True · -3 and -4 multiply to 12 and add to -7, so the brackets are correct.
  9. x squared + x + 2 · The other three split: (x + 2)(x + 3), (x - 7)(x + 3), (x + 6)(x - 3). For x squared + x + 2, no pair of integers multiplies to 2 and adds to 1.
  10. 2 · The pair is -5 and -7, so a = 5, b = 7, and b - a = 2.
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