The Discriminant and the Number of Roots · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

The Discriminant and the Number of Roots

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

  1. What is the discriminant of x squared + 10x + 25?

    Answer: ______________

  2. True or false: a discriminant of zero means the parabola touches the x-axis at exactly one point instead of crossing it twice.

    Circle one:   True   False

  3. Which of these equations has a repeated root?

    • x squared + 4x + 4
    • x squared + 4x + 5
    • x squared + 4x + 3

  4. If the discriminant of a quadratic is zero, its graph touches the x-axis once instead of crossing it twice.

    Circle one:   True   False

  5. For x squared + 6x + c = 0 to have no real roots, c = 9 works.

    Circle one:   True   False

  6. Find the value of k so that x² + 6x + k = 0 has equal roots.

    Answer: ______________

  7. Sam is finding the values of k for which x squared + kx + 4 = 0 has equal roots. He writes k = 4. What did he miss?

    • The other solution k = -4
    • The condition should be b squared > 4ac
    • The repeated root itself
    • The factor of 4 in 4ac

  8. Tom works out the discriminant of x squared + 2x + 5 as 4 - 5 = -1 and concludes there are no real roots. What is the slip in his working?

    • He subtracted in the wrong order
    • His conclusion about the number of roots is wrong
    • He squared the wrong coefficient
    • He used b squared - c instead of b squared - 4ac

  9. A quadratic equation with a = 1 has a repeated root at x = -4. What is its constant term c?

    • 8
    • -16
    • 16
    • -8

  10. x squared + bx + 16 = 0 has a repeated root, and the root is positive. What is b?

    • -8
    • 8
    • -4
    • 4
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Answer key

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

The Discriminant and the Number of Roots W1-mt_uj95V0aLVS-s1

  1. 0 · b squared is 100 and 4ac is 100, so the discriminant is 100 - 100, which is 0.
  2. True · True. A zero discriminant gives one repeated root, so the parabola's vertex sits right on the x-axis.
  3. x squared + 4x + 4 · Only x squared + 4x + 4 has a zero discriminant: 16 - 16 is 0, while the others give -4 and 4.
  4. True · Zero discriminant means one repeated root. The curve comes down to the axis, touches, and turns back: one touching point, not two crossings.
  5. False · No real roots need 36 - 4c < 0, so c must be strictly greater than 9. At c = 9 the discriminant is zero, which is a repeated root.
  6. 9 · Setting the discriminant to 0 gives k = 9.
  7. The other solution k = -4 · Setting the discriminant to zero gives k squared - 16 = 0, so k squared is 16 and k is 4 or -4. Sam only gave the positive one.
  8. He used b squared - c instead of b squared - 4ac · The correct working is 4 - 4(1)(5), which is 4 - 20, giving -16. Tom dropped the factor of 4, though his conclusion still happened to be right.
  9. 16 · A repeated root at x = -4 means the equation is (x + 4) squared, which expands to x squared + 8x + 16, so c is 16.
  10. -8 · A positive repeated root with c = 16 means the factor is (x - 4), twice, so the equation is x squared - 8x + 16 and b is -8.
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