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← Senior Mathematics Studio

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Choose the complete solution set.

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

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高一必修 第一册(A版).pdf · 2.3 · PDF 57 / printed page 50

Revisit first: Vertex form and completing the square

TOPIC 01

Roots and completeness of solutions

Choose a root method, then check reality, location and the original domain.

What you will be able to explain

  • Distinguish root values, distinct root count and root signs.
  • Separate degenerate parameter cases before using a quadratic rule.

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

Choose the complete solution set.

x(x−3)=0x(x-3)=0
  • All inputs are real unless a smaller domain is stated.
Skills and prerequisite lessons

Scope: JM01 / JM02

  1. {3}
  2. {0}
  3. {0,3}
  4. {0,−3}
Working and explanation
BUILD THE REASONING
Hint 1
Do not divide by x yet.
Hint 2
Either factor can be zero.
Worked solution
  1. Treat the first factor.
    x=0x=0
  2. Treat the second factor.
    x=3x=3
C: {0,3}.
Checks and common pitfalls: Division by x without a zero case loses a valid root.
Think first. Reveal a hint when the class is ready.
Focus on one question
Teacher preparation and assessment
40-minute lesson
MinutesFocusTeaching and student action
5Count and cancelContrast repeated roots and lost zeros with diagnostics.
8Derive ΔDerive the squared equation, marking a≠0.
10Method sortingPairs sort root-value, sum and intersection tasks.
12Case auditGroups solve the k problem and audit the linear branch.
5Repair a claimUse a counterexample to separate positive roots from real roots.
Question sequence
  • Are we finding values, a count, signs, or coordinates?
  • What assumption permits this division?
Board development
  • Count real roots through a square.
    (2ax+b)2=Δ(2ax+b)^2=\Delta
  • Partition before using a rule.
    k=1∣k≠1k=1\quad|\quad k\ne1
Expected correct and incorrect responses
  • “Two equal factors, one input” correctly separates multiplicity.
  • “k=2 only” reveals a missing linear branch.
Follow-up questions
  • Can Δ>0 coexist with a negative product of roots?
Remediation
  • Use candidate/domain/substitution columns before accepting a root.
Exit check
  • Set m=2 in (m−2)x²+x=0 and justify x=0 without a quadratic rule.
Assessment checklist and sample responses
  • 2 marks: complete branches.
    Full: k=1 and k=2; partial: only Δ=0.
  • 2 marks: preserve original restrictions.
    Full rational answer: −2 with x≠2; incorrect: ±2.
No sign-in. Work stays in this browser. Export before clearing browser data. Written reasoning is assessed with a checklist.
Curriculum and source notes ↗