Kenneth ChengMathematical modeling
← Senior Mathematics Studio

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Solve the quadratic over the complex numbers.

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

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高一必修 第二册(A版).pdf · 7.2 · PDF 82 / printed page 75

Revisit first: Concept of complex numbers

TOPIC 01

Arithmetic of complex numbers

Build understanding of arithmetic of complex numbers through definitions, contrasting cases and justified applications.

What you will be able to explain

  • Connect geometric and algebraic forms of arithmetic of complex numbers.
  • Connect representations and justify the steps, including boundary cases.
  • Explain a solution and apply the idea to a changed situation.

Try it. Leave your reasoning visible.

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

01 / Standard#Your turn

Solve the quadratic over the complex numbers.

z2−6z+10=0z^2-6z+10=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

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

  2. Calculate or simplify this relation.

    (z−3)2=−1⇒z=3±i(z-3)^2=-1\Rightarrow z=3±i

  3. Substituting either root gives zero in the original polynomial.

z=3±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.

Focus on one question

Teacher preparation and assessment

Question sequence

  • What must be true before using the main rule for arithmetic of complex numbers?
  • Which representation makes this task easier, and why?
  • Change one assumption. Does the conclusion survive?

Board plan

  • Connect geometric and algebraic forms of arithmetic of complex numbers.
  • Multiplication: Expand algebraically and replace i² with −1.
  • Division: A conjugate turns a nonzero complex denominator into its positive squared modulus.
  • Close with: conditions → representation → reasoning → check.

Anticipated thinking

  • Expected reasoning: Expand algebraically and replace i² with −1.
  • Expected reasoning: A conjugate turns a nonzero complex denominator into its positive squared modulus.
  • Expected correction: The sign from i² must not be lost.

Assessment checklist

  • 1: identify the givens and required quantity.
  • 1: choose a valid definition, representation or method.
  • 1: present connected, correct reasoning.
  • 1: check conditions and explain the result.

No sign-in. Work stays in this browser. Export before clearing browser data. Written reasoning is assessed with a checklist.

Curriculum and source notes ↗