LEARN · EXPLAIN · REVISE

Find every k giving exactly one distinct real solution, including nonquadratic cases.

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.

Examples 1–3: study a full demonstration; finish a solution from its first step; then choose a method independently before opening the solution. The remaining exercises may continue after class.

01 / Transfer#Your turn

Find every k giving exactly one distinct real solution, including nonquadratic cases.

(k-1)x^2+2x+1=0

All inputs are real unless a smaller domain is stated.

Skills and prerequisite lessons

Scope: JM01 / JM02

Working and explanation

BUILD THE REASONING

Hint 1

Separate k=1 before applying Δ.

Hint 2

For k≠1, Δ=8−4k.

Worked solution

The linear branch has one solution.
$k=1\Rightarrow x=-1/2$
The quadratic branch needs a repeated root.
$k\ne1,\quad8-4k=0\Rightarrow k=2,\quad x=-1$

k=1 or 2.

Checks and common pitfalls: A discriminant-only answer misses the linear branch.

Reasoning checklist · self / teacher assessment

The linear branch has one solution.
The quadratic branch needs a repeated root.

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

Teacher preparation and assessment

40-minute lesson

Minutes	Focus	Teaching and student action
5	Count and cancel	Contrast repeated roots and lost zeros with diagnostics.
8	Derive Δ	Derive the squared equation, marking a≠0.
10	Method sorting	Pairs sort root-value, sum and intersection tasks.
12	Case audit	Groups solve the k problem and audit the linear branch.
5	Repair a claim	Use 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.
$(2ax+b)^2=\Delta$
Partition before using a rule.
$k=1\quad|\quad k\ne1$

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.