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Algebra II · 35 minutes

Suggested grades 10–12 · Unit 2: Complex solutions and exponential models · Lesson 1 of 2

Quadratics with Complex Solutions

Extend square-root reasoning beyond real-number solutions.

Mr. S Teaches · Original starter lesson · HS-ALG2-03 · Grade range is planning guidance.

Watch, notice, discuss

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After watching

What does a negative discriminant tell you, and how can you verify a complex root?

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Learning objective

Solve real-coefficient quadratic equations with complex roots and verify a result.

Before you begin

  • Completing the square
  • Square roots

Gather your materials

  • Paper and pencil
  • Graph paper
  • Calculator for checking results

Language to know

imaginary unit
The number i defined by i²=−1.
complex number
A number a+bi with real a and b.
conjugate pair
Numbers a+bi and a−bi.

Understand the concept

A negative square is impossible for a real number, but the complex system includes i with i²=−1. Solving (x−2)²=−9 gives x−2=±3i rather than “no solution” without qualification. “No real solution” and “no complex solution” are different claims. Real-coefficient quadratics produce conjugate nonreal roots.

Teach it step by step

  1. Solve x²−4x+13=0 by completing the square: (x−2)²=−9.
  2. Use i²=−1 to express both square roots of −9.
  3. Write x=2±3i and verify one root by substitution or the completed-square form.
  4. Compare with x²−4x−5=0 to distinguish negative and positive discriminant situations.

Make the thinking visible

Worked example or model response

For x=2+3i, (x−2)²=(3i)²=−9, so the completed-square equation holds. The other root is 2−3i. In contrast x²−4x−5=(x−5)(x+1), with real roots 5 and−1.

Guided practice

Solve (x+1)²=−16 and verify one candidate.

Independent practice

1. Solve x²+6x+13=0. 2. Solve 2x²+8=0. 3. Explain the phrase “no real roots” for x²+1=0.

Quick assessment

Why is √(−9) represented by 3i rather than−3?

Answer key & assessment criteria

Answers and teaching guidance

  1. Guided: x=−1±4i; squaring x+1 gives−16.
  2. Independent 1:(x+3)²=−4, roots−3±2i. 2: x²=−4, roots±2i. 3: The complex roots are±i, but neither is real.
  3. Quick check:(−3)²=9, whereas (3i)²=−9.

Look for: Require the learner to explain the interpretation and check, not only state a result. Distinguish an exact value from a rounded approximation and attend to the stated domain.

Make it work for your learner

More support

Use a short square table for real and imaginary multiples before completing the square.

Extension challenge

Choose real coefficients producing roots 4±i and verify your equation.

Verified references · Selected contributions

Standards connections

Subject-content standards

These records identify the part of a subject standard practiced here. Course placement is instructional guidance; the source’s category or band remains authoritative.

Contributes to Common Core standard

HSN.CN.C.7

Common Core · High school · Number and Quantity

Regular high school mathematics standard

Solve real-coefficient quadratics whose roots are nonreal complex numbers.

Read the official standard ↗

This lesson contributes selected skills; broader applications and full standard coverage require further instruction.

Verified 2026-11-04. Selected contribution, not full attainment.