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Complex Solutions of Quadratics

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

Quadratics with negative discriminants have complex solutions. For real-coefficient quadratics, nonreal complex roots occur in conjugate pairs.

What you'll learn

Why it matters: When a parabola never crosses the x-axis, the equation still has solutions, but they are complex. This idea supports engineering, signal, and polynomial work where no real intercept does not mean no algebraic answer.

Worked example

Problem. Solve x^2 + 25 = 0.

  1. x^2 = -25.
  2. Take square roots.
  3. x = +/-5i.

Answer: x = 5i or x = -5i

Practice problems

1. Solve x^2 + 16 = 0.

Choices: x = 4i or x = -4i · x = 4 or x = -4 · x = 16i · No solution

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 or x = -4i

2. Find the discriminant of x^2 + 2x + 10 = 0.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the discriminant of x^2 + 2x + 10 = 0.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. 4 - 40 = -36.
  4. Check the result by substituting or estimating: the response should match -36 and make sense in the original problem.

Answer: -36

3. If 6 - 2i is a root of a real quadratic, the other root is...

Choices: 6 + 2i · -6 - 2i · -6 + 2i · 2 - 6i

Show solution
  1. Challenge: First identify exactly what the question is asking: If 6 - 2i 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. Use the conjugate.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 6 + 2i

4. Solve x^2 + 25 = 0.

Choices: x = 5i or x = -5i · x = 5 or x = -5 · x = 25i · No solution

Show solution
  1. Imaginary Roots: First identify exactly what the question is asking: Solve x^2 + 25 = 0.
  2. For complex numbers, use i^2 = -1 and combine real parts with real parts and imaginary parts with imaginary parts.
  3. x^2 = -25.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: x = 5i or x = -5i

5. What is the discriminant of x^2 + 4x + 13?

Show solution
  1. Discriminant: First identify exactly what the question is asking: What is the discriminant of x^2 + 4x + 13?
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. b^2 - 4ac = 16 - 52.
  4. Check the result by substituting or estimating: the response should match -36 and make sense in the original problem.

Answer: -36

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