Factorising quadratics with a leading coefficient · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Factorising quadratics with a leading coefficient

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

  1. Split the middle term of 4x squared + 13x + 3. The product a times c is 12, and the pair adds to 13. Writing the larger part first, what is the coefficient of the first x term in the split form?

    Answer: ______________

  2. True or false: (x + 3)(2x + 1) has an x squared term with coefficient 1.

    Circle one:   True   False

  3. A proposed factorisation agrees with the original quadratic at x = 0. That check alone proves the factorisation is correct.

    Circle one:   True   False

  4. The middle term of 6x² + bx + 3 is split using the coefficients 9 and 2. What is b?

    Answer: ______________

  5. What is the greatest common factor of the terms 6x² and 18x?

    • 3x
    • 6x
    • 6x²
    • 18x

  6. Factorise 4x squared - 4x - 3.

    • (2x + 3)(2x - 1)
    • (2x + 1)(2x - 3)
    • (4x + 3)(x - 1)

  7. 4x squared + 4x + 1 is a perfect square, (2x + 1) squared.

    Circle one:   True   False

  8. Which quadratic does (2x + 3)(4x + 1) expand to?

    • 8x squared + 14x + 3
    • 8x squared + 12x + 3
    • 6x squared + 4x + 3

  9. A rectangle has area 6x squared + 17x + 7 square units, and one side is 2x + 1 units. What is the other side?

    • 2x + 7
    • 3x + 1
    • 6x + 7
    • 3x + 7

  10. Fully factorise -3x² - 15x - 18.

    • -3(x - 2)(x - 3)
    • 3(x + 2)(x + 3)
    • -3(x + 1)(x + 6)
    • -3(x + 2)(x + 3)
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Answer key

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Factorising quadratics with a leading coefficient W1-mt_IdKGCzbH07-s1

  1. 12 · The pair 12 and 1 splits the middle term into 12x and x, so the first x coefficient is 12.
  2. False · The leading term comes from x times 2x, which is 2x². Its coefficient is 2, so the statement is false. In general, multiply the two brackets' x coefficients to find the leading coefficient.
  3. False · At x = 0, only the constant is checked. Different quadratics can have the same constant. Expand and compare every coefficient.
  4. 11 · The split is 9x + 2x = 11x, so b = 11.
  5. 6x · The coefficients share a greatest factor of 6, and both terms contain at least one x. The greatest common factor is 6x.
  6. (2x + 1)(2x - 3) · The product is -12, and the pair -6 and 2 adds to -4, so the split 4x squared - 6x + 2x - 3 groups to (2x + 1)(2x - 3).
  7. True · The product is 4 and the pair 2 and 2 adds to 4, so the split groups to (2x + 1)(2x + 1).
  8. 8x squared + 14x + 3 · The cross terms are 2x and 12x, which collect to 14x, and the constant is 3.
  9. 3x + 7 · Area is side times side, so factorising the area gives the two sides. The factorisation is (2x + 1)(3x + 7), so the other side is 3x + 7.
  10. -3(x + 2)(x + 3) · Take out -3 to get -3(x² + 5x + 6). The pair 2 and 3 has product 6 and sum 5, so the result is -3(x + 2)(x + 3).
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