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Quadratic equations with complex roots

Edexcel IAL Further Mathematics · WFM01 Further Pure Mathematics 1

Every quadratic with real coefficients has two roots. When the discriminant is negative they are a complex conjugate pair, and you can work backwards from one root to the whole equation.

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Negative discriminant

For az2 + bz + c = 0 with a, b, c real: if b2 - 4ac < 0, the roots are z = (-b ± i sqrt(4ac - b2))/(2a), a conjugate pair p ± qi.

You can also complete the square: z2 - 6z + 34 = (z - 3)2 + 25 = 0, so z - 3 = ±5i and z = 3 ± 5i.

From roots to equation

If the roots are α and α*, the quadratic is (z - α)(z - α*) = z2 - (α + α*)z + αα*.

α + α* = 2 Re(α) and αα* = |α|2, both real. For α = 4 - 3i: z2 - 8z + 25 = 0.

Watch out

Keep the ± and the square root in the formula; a common error is to lose the i. With a leading coefficient a ≠ 1, divide the whole of the top by 2a: (6 ± 2i)/4 = 3/2 ± i/2.

Worked example

Solve 2z2 - 6z + 5 = 0.

  1. b2 - 4ac = 36 - 40 = -4, so sqrt(-4) = 2i.
  2. z = (6 ± 2i)/4.

Answer: z = 3/2 + i/2 or z = 3/2 - i/2

Key idea: Negative discriminant gives roots p ± qi. A quadratic with roots α and α* is z2 - 2Re(α)z + |α|2 = 0.

Check yourself

  1. Solve z2 - 6z + 34 = 0. Give the imaginary part of the root with positive imaginary part.
    Show answer

    5. (z - 3)2 + 25 = 0 so z = 3 ± 5i. The imaginary part is 5.

  2. Solve 2z2 - 6z + 5 = 0. Find the real part of the roots. Give your answer as a fraction in its simplest form.
    Show answer

    3/2. z = (6 ± sqrt(36 - 40))/4 = (6 ± 2i)/4 = 3/2 ± i/2.

  3. Find the quadratic z2 + bz + c, with real b and c, that has 4 - 3i as a root.
    Show answer

    z2 - 8z + 25. The other root is 4 + 3i. Sum = 8 and product = 16 + 9 = 25, so z2 - 8z + 25.

  4. z2 + bz + c = 0, with b and c real, has a root 2 + i. Find c.
    Show answer

    5. The roots are 2 ± i. c is the product of the roots: (2 + i)(2 - i) = 4 + 1 = 5.

  5. z2 + bz + c = 0, with b and c real, has a root 2 + i. Find b.
    Show answer

    -4. The sum of the roots is (2 + i) + (2 - i) = 4 and the sum is -b, so b = -4.

  6. The roots of z2 - 4z + 13 = 0 are plotted on an Argand diagram. Find their distance from O, to 3 decimal places.
    Show answer

    3.606. z = 2 ± 3i and |2 ± 3i| = sqrt(13) = 3.606.

  7. A quadratic with real coefficients has discriminant -20. What can you say about its roots?
    • They are two equal real roots
    • They are two distinct real roots
    • They are two non-real complex conjugates
    • There are no roots of any kind
    Show answer

    They are two non-real complex conjugates. A negative discriminant means the square root is imaginary, giving p ± qi.

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Last updated 8 November 2026. Spotted a mistake? Tell us.