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Refute “Δ>0 means two positive roots”, then give necessary and sufficient conditions for two distinct positive roots.

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

Refute “Δ>0 means two positive roots”, then give necessary and sufficient conditions for two distinct positive roots.

x^2+bx+c=0

The polynomial is monic with real coefficients.

Skills and prerequisite lessons

Scope: JM01 / JM02

Working and explanation

BUILD THE REASONING

Hint 1

Separate realness from signs.

Hint 2

After Δ>0, inspect the sum −b and product c.

Worked solution

A counterexample invalidates the proposed condition.
$x^2-1=0\Rightarrow x=\pm1$
Positive product gives the same sign; positive sum makes it positive.
$b^2-4c>0,\quad b<0,\quad c>0$

Exactly Δ>0, b<0 and c>0.

Checks and common pitfalls: Positive roots imply all three conditions; conversely these conditions force two same-sign real roots with positive sum.

Reasoning checklist · self / teacher assessment

A counterexample invalidates the proposed condition.
Positive product gives the same sign; positive sum makes it positive.

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.