Solve Quadratic Equations with Complex Solutions

Lesson 7 of 9: Complex Number System

In this lesson:

• Identify complex solutions using the discriminant
• Apply the quadratic formula to get complex solutions
• Recognize that complex solutions come in conjugate pairs
• Verify solutions by substitution

What You Will Learn Today

1. Use the discriminant to determine when solutions are complex
2. Apply the quadratic formula and write complex solutions in form
3. Recognize that complex solutions always appear as conjugate pairs
4. Verify complex solutions by substitution

Recall: Quadratic Formula and Discriminant

• Formula: for
• Discriminant : two real solutions
• Discriminant : one repeated real solution

What did a negative discriminant mean before this lesson?

The Upgrade: Complex Solutions, Not "No Solution"

Old framing (real numbers only):
: discriminant → "No solution."

New framing (complex numbers):
Discriminant → Two complex solutions. Find them.

A negative discriminant is a signal, not a dead end.

Three Cases of the Discriminant

Negative Discriminant: Complex Solutions Arise

When (real coefficients):

• No real number is a solution
• Two complex conjugate solutions exist: and

"No real solution" "no solution."

Why : A One-Line Proof

Claim:

Check: ✓

General rule: for

. Check:

Five-Step Template for Complex Quadratic Solutions

1. Compute discriminant (must be negative)
2. Apply:
3. Convert:
4. Divide both real and imaginary parts by
5. Write the conjugate pair: and

Worked Example One: Negative Discriminant

Step 1: , , ; discriminant

Steps 2–4:

Step 5: Solutions: and

Worked Example Two: Different Sign Pattern

Step 1: , , ; discriminant

Steps 2–4:

Step 5: Solutions: and

Conjugate Pairs: Both Solutions Always Appear

For real-coefficient quadratics:

The produces both and .

If is a solution, then is also a solution.

Never list only one of a conjugate pair.

Quick Check: Two Complex Quadratics

1. Find both solutions of
2. Find both solutions of

For each: compute the discriminant, apply the formula, write the conjugate pair.

Verification: Substitution Works for Complex Roots

To verify solves : substitute and simplify to .

Why it works: If the algebra giving was correct, substituting must produce — that is the definition of "solution."

Complex arithmetic (CN.A.2) handles all the steps.

Verify a Complex Solution by Substitution

Substitute into :

Real parts: . Imaginary parts: .

Completing the Square: Same Complex Results

Solve by completing the square:

Same result as the quadratic formula.

Vieta's Formulas: Sum and Product Are Real

For with roots and :

Sum: ✓

Product: ✓

Conjugate pairs always give real sum and real product.

Find the Two Errors in This Work

Solving :

1. Student A: , so or .

2. Student B: .

Correct both errors.

Three Problems With No Scaffolding

1. Classify: Does have real or complex solutions?

2. Solve: Find both solutions of in form.

3. Build: Write a quadratic with real coefficients whose solutions are .

Watch for These Three Common Errors

1. sqrt error: , not (check: ✓)

2. Division error: , not

3. Missing conjugate: and — list both

What You Now Know About Complex Quadratics

✓ Negative discriminant → two complex conjugate solutions
✓ : imaginary, never negative real
✓ Formula gives both: and
✓ Verify: substitute into the equation, get

Divide both real and imaginary parts by

Next: Polynomial Identities and Complex Numbers

In HSN.CN.C.8, you'll factor expressions like :

The roots of are — the complex conjugate pair from this lesson applied to factoring.