LEARN · EXPLAIN · REVISE

Roots and completeness of solutions

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

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.

Count through a square

A count question may not require individual roots.

The discriminant controls distinct real root count for a genuine quadratic.

\Delta=b^2-4ac

Complete the equation as a square.
$(2ax+b)^2=\Delta$
Read the three possibilities.
$\Delta>0:2;\quad\Delta=0:1;\quad\Delta<0:0$

When it applies

a≠0; count distinct real inputs.

Counterexample: A linear equation can have one root even though the quadratic rule is inapplicable.

Connect the square equation with x-axis intersections.

Keep candidates and restrictions separate

Root count alone cannot tell whether roots are positive or physically possible.

Cancellation and modelling can exclude candidates, so test the original domain.

r+s=-b/a,\quad rs=c/a

Factor forms reveal signs and restrictions.
$x(x-3)=0$
Retain the zero case before division.
$x=0\text{ or }3$

When it applies

Never divide by a possibly zero expression without a case split.

Counterexample: Δ>0 does not imply two positive roots: x²−1 has roots ±1.

Use a candidate table: domain, substitution, accepted or rejected.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: As c crosses the tangency level, how does the root count change?

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

Equation type: quadratic. Real zeros: 0, 2. Discriminant = 4.

Numerical data for the diagram; printable
Root	Substitution residual
0	0
2	0

Explain: Compare crossings with Δ and set a=0 to test degeneration.

Transfer: Choose Δ>0 with both roots negative.

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 / Foundation#Worked example

Solve and check both roots.

x^2-5x+6=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

Seek product 6 and sum −5.

Hint 2

Use −2 and −3.

Worked solution

Factor the polynomial.
$(x-2)(x-3)=0$
Apply zero product and substitute.
$x=2,3;\quad4-10+6=9-15+6=0$

x=2 or 3.

Checks and common pitfalls: Both factors must be considered.

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

02 / Standard#Worked example

Count distinct real roots without approximating them.

2x^2+4x+5=0

First step provided — complete the reasoning:Confirm a≠0 and calculate.

\Delta=4^2-4(2)(5)=-24

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

Use Δ for a count question.

Hint 2

Compare 16 with 40.

Worked solution

Confirm a≠0 and calculate.
$\Delta=4^2-4(2)(5)=-24$
A real square cannot be negative.
$N=0$

Checks and common pitfalls: This concerns real roots; complex roots do not count here.

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

03 / Transfer#Worked example

Find every k giving tangency to the x-axis and the contact input.

y=x^2-2kx+k+2

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

Tangency means one repeated root.

Hint 2

Set k²−k−2=0.

Worked solution

The leading coefficient is always nonzero.
$\Delta=4(k^2-k-2)=0$
Solve the parameter equation and recover the axis.
$k=2,-1;\quad x=k$

k=2 at x=2, or k=−1 at x=−1.

Checks and common pitfalls: Both the constant term and axis change with k.

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

04 / Foundation#Your turn

Count distinct real roots.

(x-4)^2=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

Zero square forces zero base.

Hint 2

Repeated factors give the same value.

Worked solution

Solve the base equation.
$x=4$
Count values, not factors.
$N=1$

Checks and common pitfalls: Multiplicity two is not two distinct inputs.

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

05 / Foundation#Your turn

Find the root sum without finding each root.

x^2-7x+4=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

Expand (x−r)(x−s).

Hint 2

The x coefficient is −(r+s).

Worked solution

Compare coefficients.
$-(r+s)=-7$
Real roots exist since the discriminant is positive.
$r+s=7,\quad\Delta=33$

Checks and common pitfalls: The sum can be simple even when each root is irrational.

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

06 / Foundation#Your turn

Solve over the real numbers.

x^2=7

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

Include both signs.

Hint 2

Both signed square roots square to 7.

Worked solution

Take both real branches.
$x=\pm\sqrt7$
Check both signs by squaring.
$(\sqrt7)^2=(-\sqrt7)^2=7$

x=±√7.

Checks and common pitfalls: The principal square-root symbol alone is nonnegative.

Reasoning checklist · self / teacher assessment

Take both real branches.
Check both signs by squaring.

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

07 / Foundation#Your turn

Give the positive root.

2(x-3)(x+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

The factor 2 never vanishes.

Hint 2

Solve each linear factor.

Worked solution

List both roots.
$x=3,-1$
Select the positive candidate.
$x=3$

Checks and common pitfalls: Apply the sign condition after forming the complete root set.

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

08 / Standard#Your turn

Find all intersection points.

y=x^2,\quad y=2x+3

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

Equate the heights.

Hint 2

Solve x²−2x−3=0.

Worked solution

Find the intersection inputs.
$(x-3)(x+1)=0$
Recover the outputs from either graph.
$(x,y)=(3,9),(-1,1)$

(−1,1) and (3,9).

Checks and common pitfalls: An intersection includes the output as well as the root input.

Reasoning checklist · self / teacher assessment

Find the intersection inputs.
Recover the outputs from either graph.

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

09 / Standard#Your turn

One root is 3. Find the other.

2x^2-4x-6=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

Use sum or product.

Hint 2

The root sum is 2.

Worked solution

Read the coefficient ratio.
$r+s=-(-4)/2=2$
Subtract and check the product.
$s=-1,\quad3(-1)=-6/2$

−1.

Checks and common pitfalls: Do not forget the leading coefficient in the ratios.

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

10 / Standard#Your turn

Solve, recording excluded inputs.

\frac{x^2-4}{x-2}=0

x≠2.

Skills and prerequisite lessons

Scope: JM01 / JM02

Working and explanation

BUILD THE REASONING

Hint 1

Record the original denominator restriction first.

Hint 2

Cancellation leaves x+2=0 with x≠2.

Worked solution

Preserve the domain while cancelling.
$x\ne2,\quad(x^2-4)/(x-2)=x+2$
Solve and verify the denominator is nonzero.
$x=-2,\quad x-2=-4$

Only x=−2; x=2 is excluded.

Checks and common pitfalls: The cancelled factor zero was never allowed.

Reasoning checklist · self / teacher assessment

Preserve the domain while cancelling.
Solve and verify the denominator is nonzero.

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

11 / Standard#Your turn

A rectangle has sides x and x+3, area 10. Find x.

x(x+3)=10,\quad x>0

Both sides must be positive.

Skills and prerequisite lessons

Scope: JM01 / JM02

Working and explanation

BUILD THE REASONING

Hint 1

Form a quadratic in standard form.

Hint 2

Factor x²+3x−10.

Worked solution

Find algebraic candidates.
$(x+5)(x-2)=0$
Reject the negative side length.
$x=2$

2 length units.

Checks and common pitfalls: The negative root is algebraically valid but fails the model.

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

12 / 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.

13 / 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.

End-of-lesson check and correction

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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.