The Quadratic Formula · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

The Quadratic Formula

Mathematics · Algebra · ages 15-16
Name ______________________   Date ____________

  1. Use the quadratic formula on x² - 5x + 6 = 0. What are the roots?

    • x = -2 or x = -3
    • x = 1 or x = 6
    • x = 2 or x = 3
    • x = 5 or x = 1

  2. Which equation is the formula the right tool for, because factorising with whole numbers will not work?

    • x squared - 4x - 2 = 0
    • x squared - 5x + 6 = 0
    • x squared - 7x + 12 = 0
    • x squared + 2x = 0

  3. Ben tries to factorise x squared - 6x + 2 = 0 with whole numbers and cannot. Why does the formula still work?

    • It works for any quadratic, whether it factorises or not
    • It only works when the discriminant is a square number
    • It needs the equation to factorise first
    • It only finds one of the two roots

  4. In x squared + 5x - 8 = 0, what is -b, the number that goes on top of the formula?

    • 5
    • -8
    • 8
    • -5

  5. Solve 2x squared - 3x - 1 = 0 with the formula. What is the exact form of the larger root?

    • (3 plus root 17) / 2
    • (3 plus root 17) / 4
    • (3 plus root 34) / 4
    • (3 plus root 17) / 8

  6. A learner substituted into the formula and got 81 under the root sign. If b squared was 49, what was 4ac?

    • 32
    • 113
    • -32
    • -113

  7. Solve 2x² + 8x + 6 = 0 with the quadratic formula. What is the larger root?

    Answer: ______________

  8. Solve x squared - 2x - 5 = 0 with the formula. Which choice gives both roots exactly?

    • 1 plus or minus root 24
    • 1 plus or minus 6
    • -1 plus or minus root 6
    • 1 plus or minus root 6

  9. Dev solves 3x squared - 4x - 1 = 0 with the formula and writes x = (4 plus/minus root 4) / 6. His substitution is correct.

    Circle one:   True   False

  10. Solve x squared - 5x + 3 = 0 with the formula. Give the larger root to 2 decimal places.

    Answer: ______________

LightMySky · lightmysky.comW1-mt_C-4brnRYHY-s1

Answer key

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

The Quadratic Formula W1-mt_C-4brnRYHY-s1

  1. x = 2 or x = 3 · With a = 1, b = -5, and c = 6, the formula gives (5 plus or minus 1)/2, which is 2 or 3.
  2. x squared - 4x - 2 = 0 · x squared - 4x - 2 has no whole-number pair multiplying to -2 and adding to -4, so factorising fails and the formula is the right tool. The other three factorise: (x - 2)(x - 3), (x - 3)(x - 4), and x(x + 2).
  3. It works for any quadratic, whether it factorises or not · The formula is built from completing the square on the general quadratic, so it solves every quadratic with real roots. No factorising is needed first.
  4. -5 · b is 5, and the formula uses -b, which flips the sign to -5.
  5. (3 plus root 17) / 4 · The formula gives (3 plus/minus the root of 17) / 4, and 17 is not a square, so the larger root stays as (3 plus root 17) / 4.
  6. -32 · The number under the root is b squared - 4ac, so 49 - 4ac = 81, which gives 4ac = -32.
  7. -1 · With a = 2, b = 8, and c = 6, the roots are -3 and -1, so the larger root is -1.
  8. 1 plus or minus root 6 · The formula gives (2 plus/minus the root of 24) / 2, the root of 24 simplifies to 2 root 6, and dividing by 2 leaves 1 plus or minus root 6.
  9. False · The top and bottom are right, but 4ac is -12, so b squared - 4ac is 16 + 12, which is 28, not 4.
  10. 4.3 · The formula gives (5 plus/minus the root of 13) / 2, the root of 13 is about 3.606, so the larger root is 8.606 / 2, about 4.30.
Worksheet · LightMySky