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Solve the quadratic over the complex numbers.

Read the idea, work independently, then explain what changed.

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TOPIC 01

Complex numbers: mixed assessment

A mixed assessment: identify the method, justify it and revise your reasoning.

What you will be able to explain

Connect the chapter skills without relying on the order of the exercises.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Your turn

Solve the quadratic over the complex numbers.

z^2-16z+65=0

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Expand with i²=−1 and use a conjugate to remove a complex denominator.

Hint 2

Complete the square.

Worked solution

Expand with i²=−1 and use a conjugate to remove a complex denominator.
Calculate or simplify this relation.
$(z-8)^2=-1\Rightarrow z=8±i$
Substituting either root gives zero in the original polynomial.

z=8±i.

Checks and common pitfalls: Substituting either root gives zero in the original polynomial.

Reasoning checklist · self / teacher assessment

State the relevant definition, condition or model.
Show a valid calculation, proof or counterexample.
Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

Teacher preparation and assessment

Question sequence

Ask students to name the relevant condition before calculating.

Board plan

Compare valid methods and annotate their conditions.

Anticipated thinking

A correct final value may still hide a missing assumption.

Assessment checklist

Check the method, conditions, reasoning and interpretation separately.