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. Expanding (x + 4)(x + 3) gives x squared + 7x + 12, so it is a valid check for a factorisation of x squared + 7x + 12.

    Circle one:   True   False

  2. The pair for x squared + 9x + 20 multiplies to 20 and adds to 9. Which pair fits?

    • 4 and 5
    • 2 and 10
    • 1 and 20

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

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

  4. Factorise x squared + 6x + 8.

    • (x + 1)(x + 8)
    • (x + 2)(x + 4)
    • (x - 2)(x - 4)

  5. Factorise x squared - 5x - 14.

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

  6. A friend claims x squared + 6x + 8 = (x + 6)(x + 8). How do you respond?

    • It is correct, because 6 and 8 add to 6
    • It is correct, because 6 and 8 multiply to 8
    • Expand the brackets: the x term would be 14x, not 6x

  7. Two brackets expand to x squared - 3x + 2. What were the brackets?

    • (x + 1)(x - 2)
    • (x - 3)(x + 1)
    • (x - 1)(x - 2)

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

    Circle one:   True   False

  9. If a pair multiplies to c but does not add to b, it can still go into the brackets.

    Circle one:   True   False

  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 · The expansion lands exactly on the original quadratic, which is what a check is for.
  2. 4 and 5 · 4 and 5 multiply to 20 and add to 9, giving (x + 4)(x + 5).
  3. -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).
  4. (x + 2)(x + 4) · 2 and 4 multiply to 8 and add to 6, so the brackets are (x + 2)(x + 4).
  5. (x - 7)(x + 2) · The pair multiplies to -14 and adds to -5, which is -7 and 2, giving (x - 7)(x + 2).
  6. Expand the brackets: the x term would be 14x, not 6x · Expanding (x + 6)(x + 8) gives x squared + 14x + 48, which is not the original, so the claim fails the check.
  7. (x - 1)(x - 2) · The pair multiplies to 2 and adds to -3, which is -1 and -2, so the brackets are (x - 1)(x - 2).
  8. True · -3 and -4 multiply to 12 and add to -7, so the brackets are correct.
  9. False · Both jobs must be done. A pair that misses the sum gives the wrong x term when expanded.
  10. 2 · The pair is -5 and -7, so a = 5, b = 7, and b - a = 2.
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