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Complex Solutions of Quadratics

A free Algebra II lesson from the “Complex Numbers” unit, with a worked example and practice problems including step-by-step solutions.

When a quadratic's discriminant is negative, it has no real roots but two complex roots. Complex roots for real-coefficient quadratics come in conjugate pairs.

What you'll learn

Why it matters: When a parabola never touches the x-axis, its zeros are a complex conjugate pair. Engineering control systems, oscillation modeling, and signal filters all run into this case, and the quadratic formula still gives the answer — it just produces complex roots.

Worked example

Problem. Solve x^2 + 9 = 0.

  1. Subtract 9 to get x^2 = -9.
  2. Take square roots.
  3. x = +/- sqrt(-9) = +/- 3i.

Answer: x = 3i and x = -3i

Practice problems

1. Solve x^2 + 16 = 0.

Choices: x = 4i and x = -4i · x = 4 and x = -4 · x = 16i · No solutions

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve x^2 + 16 = 0.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. x^2 = -16.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: x = 4i and x = -4i

2. Find the discriminant of x^2 + 4x + 8 = 0.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find the discriminant of x^2 + 4x + 8 = 0.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. b^2 - 4ac = 16 - 32.
  4. Check the result by substituting or estimating: the response should match -16 and make sense in the original problem.

Answer: -16

3. A discriminant of -20 means the solutions are...

Choices: Complex · Two real · One repeated real · Linear

Show solution
  1. Core Practice: First identify exactly what the question is asking: A discriminant of -20 means the solutions are...
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. Negative discriminants give complex roots.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Complex

4. If 2 + 5i is a root of a real quadratic, the other root is...

Choices: 2 - 5i · -2 + 5i · -2 - 5i · 5 + 2i

Show solution
  1. Challenge: First identify exactly what the question is asking: If 2 + 5i is a root of a real quadratic, the other root is...
  2. For quadratics, track the zeros, vertex, or coefficients so the algebra matches the graph feature being asked about.
  3. Complex roots come in conjugate pairs.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 2 - 5i

5. Find the discriminant of x^2 - 6x + 13 = 0.

Show solution
  1. Mixed Review: First identify exactly what the question is asking: Find the discriminant of x^2 - 6x + 13 = 0.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Use b^2 - 4ac.
  4. (-6)^2 - 4(1)(13) = 36 - 52.
  5. The discriminant is -16, so the roots are complex.
  6. Check the result by substituting or estimating: the response should match -16 and make sense in the original problem.

Answer: -16

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