Mathematics ยท WMA11 (P1)

The discriminant

The discriminant tells you how many real roots a quadratic has without solving it.

What it is

For ax2 + bx + c = 0, the discriminant is b2 - 4ac. It is the part under the square root in the quadratic formula.

What it tells you

Finding an unknown

Questions often say 'has equal roots' or 'has no real roots'. Write the discriminant in terms of the unknown, then set it = 0 or < 0 and solve.

Worked example

x2 + kx + 9 = 0 has equal roots. Find the possible values of k.

  1. Equal roots, so b2 - 4ac = 0.
  2. k2 - 4(1)(9) = 0
  3. k2 = 36

Answer: k = 6 or k = -6

Key idea: b2 - 4ac > 0: two real roots; = 0: one repeated root; < 0: no real roots.

Check yourself

  1. Find the discriminant of 2x2 + 3x - 5 = 0.
    Show answer

    49. b2 - 4ac = 9 - 4(2)(-5) = 9 + 40 = 49.

  2. How many real roots does x2 - 4x + 7 = 0 have?
    • One
    • Two
    • Three
    • None
    Show answer

    None. b2 - 4ac = 16 - 28 = -12 < 0, so no real roots.

  3. The discriminant of a quadratic is 0. What does its graph do?
    • Crosses the x-axis twice
    • Touches the x-axis at one point
    • Never meets the x-axis
    • Crosses the y-axis twice
    Show answer

    Touches the x-axis at one point. A zero discriminant means one repeated root, so the curve just touches the x-axis.

  4. x2 + 6x + c = 0 has equal roots. Find c.
    Show answer

    9. 36 - 4c = 0, so c = 9.

  5. kx2 + 4x + 1 = 0 has two distinct real roots. Which condition must k satisfy (k is not 0)?
    • k < 4
    • k > 4
    • k = 4
    • k < -4
    Show answer

    k < 4. 16 - 4k > 0 gives k < 4.

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Sources

Last updated 7 November 2026. Spotted a mistake? Email alevel@lifereadysociety.com.