LEARN · EXPLAIN · REVISE

Quadratic functions, equations and inequalities: mixed assessment

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

TOPIC 01

Quadratic functions, equations and inequalities: mixed assessment

A mixed assessment: identify the method, justify it and revise your reasoning.

What you will be able to explain

Connect the chapter skills without relying on the order of the exercises.

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

Solve and justify the sign direction.

-7x>56

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Track signs and legal operations before transforming an inequality.

Hint 2

Divide by a negative number.

Worked solution

Track signs and legal operations before transforming an inequality.
Calculate or simplify this relation.
$x<56/(-7)=-8$
A negative multiplier reverses order.

x<-8.

Checks and common pitfalls: A negative multiplier reverses order.

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.

02 / Foundation#Your turn

Positive a,b have sum 14; maximize their product.

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Check nonnegativity, apply AM–GM and verify attainable equality.

Hint 2

Fix the arithmetic mean.

Worked solution

Check nonnegativity, apply AM–GM and verify attainable equality.
Calculate or simplify this relation.
$ab\le((a+b)/2)^2=49;\quad a=b=7$
The equal pair satisfies the required sum.

The requested value is 49.

Checks and common pitfalls: The equal pair satisfies the required sum.

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

03 / Foundation#Your turn

Calculate the discriminant and interpret its sign.

x^2-14x+50=0

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Find the roots or vertex, then check signs and endpoints.

Hint 2

Use the full coefficient of x.

Worked solution

Find the roots or vertex, then check signs and endpoints.
Calculate or simplify this relation.
$\Delta=(-14)^2-4(50)=-4$
A negative discriminant means no real root.

The requested value is -4.

Checks and common pitfalls: A negative discriminant means no real root.

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

04 / Foundation#Your turn

Give a counterexample to squaring as a general order-preserving rule.

-9<-8

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Track signs and legal operations before transforming an inequality.

Hint 2

Compare absolute values.

Worked solution

Track signs and legal operations before transforming an inequality.
Calculate or simplify this relation.
$(-9)^2=81>64=(-8)^2$
Squaring is increasing only on nonnegative inputs.

Their squares satisfy 81>64.

Checks and common pitfalls: Squaring is increasing only on nonnegative inputs.

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.

05 / Foundation#Your turn

Does the function attain a minimum on this domain?

x+\frac{64}x,\quad x>8

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Check nonnegativity, apply AM–GM and verify attainable equality.

Hint 2

The equality point is excluded.

Worked solution

Check nonnegativity, apply AM–GM and verify attainable equality.
Calculate or simplify this relation.
$x+\frac{64}x-16=\frac{(x-8)^2}{x}>0$
Calculate or simplify this relation.
$x\to8^{+}\Rightarrow x+\frac{64}x\to16$
An infimum need not be a minimum.

No; its infimum is 16, approached but not attained.

Checks and common pitfalls: An infimum need not be a minimum.

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.

06 / Foundation#Your turn

Find the length of the x-interval where the line is above or on the parabola.

y=(20)x-96,\quad y=x^2

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Find the roots or vertex, then check signs and endpoints.

Hint 2

Compare their vertical difference.

Worked solution

Find the roots or vertex, then check signs and endpoints.
Calculate or simplify this relation.
$(x-8)(x-12)\le0;\quad (12)-8=4$
Interval length is not the count of integer points.

The requested value is 4.

Checks and common pitfalls: Interval length is not the count of integer points.

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

07 / Standard#Your turn

Solve both parts and justify the conditions used. Part A: Explain how the sign of a affects multiplying x<b by a. Part B: For which m is this positive for every real x?

\begin{gathered}\text{A: }x<b\\\text{B: }x^2-20x+m\end{gathered}

Use the domain, units and sampling assumptions stated in the question.
Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

A: Track signs and legal operations before transforming an inequality.

Hint 2

B: Find the roots or vertex, then check signs and endpoints.

Worked solution

Part A reasoning
Track signs and legal operations before transforming an inequality.
Calculate or simplify this relation.
$ax-ab=a(x-b)$
The zero case is different from either strict order.
Part B reasoning
Find the roots or vertex, then check signs and endpoints.
Calculate or simplify this relation.
$x^2-20x+m=(x-10)^2+m-100$
Strict positivity requires a strictly positive minimum.

A: If a>0 then ax<ab; if a<0 then ax>ab; if a=0 the results are equal. B: m>100.

Checks and common pitfalls: The zero case is different from either strict order. Strict positivity requires a strictly positive minimum.

Reasoning checklist · self / teacher assessment

Solve part A with its stated restrictions.
Solve part B using an appropriate representation.
Give the reasoning and check conditions in both parts.

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

08 / Standard#Your turn

Find the minimum for x>0.

x+\frac{81}x

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Check nonnegativity, apply AM–GM and verify attainable equality.

Hint 2

The two positive terms have constant product.

Worked solution

Check nonnegativity, apply AM–GM and verify attainable equality.
Calculate or simplify this relation.
$x+\frac{81}x\ge2\sqrt{81}=18;\quad x=9$
Equal terms give an admissible equality case.

The requested value is 18.

Checks and common pitfalls: Equal terms give an admissible equality case.

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

09 / Standard#Your turn

Count the integer solutions.

(x+9)(x-11)<0

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Find the roots or vertex, then check signs and endpoints.

Hint 2

Exclude the two integer roots.

Worked solution

Find the roots or vertex, then check signs and endpoints.
Calculate or simplify this relation.
$-9<x<11;\quad (10)-(-8)+1=19$
Strict comparisons exclude equality.

The requested value is 19.

Checks and common pitfalls: Strict comparisons exclude equality.

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

10 / Standard#Your turn

Find the least integer greater than every admissible sum a+b.

a<10,\ b<12

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Track signs and legal operations before transforming an inequality.

Hint 2

Add the upper bounds.

Worked solution

Track signs and legal operations before transforming an inequality.
Calculate or simplify this relation.
$a+b<22$
Sums approach this bound arbitrarily closely, so no smaller integer works.

The requested value is 22.

Checks and common pitfalls: Sums approach this bound arbitrarily closely, so no smaller integer works.

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

11 / Standard#Your turn

Minimize the expression over positive x.

2x+\frac{200}x

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Check nonnegativity, apply AM–GM and verify attainable equality.

Hint 2

Include the coefficient two in the product.

Worked solution

Check nonnegativity, apply AM–GM and verify attainable equality.
Calculate or simplify this relation.
$2x+\frac{200}x\ge2\sqrt{400}=40;\quad x=10$
Equality concerns the complete two terms, not just x and 1/x.

The requested value is 40.

Checks and common pitfalls: Equality concerns the complete two terms, not just x and 1/x.

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

12 / Standard#Your turn

For which m is this positive for every real x?

x^2-20x+m

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Find the roots or vertex, then check signs and endpoints.

Hint 2

Its minimum occurs when the square vanishes.

Worked solution

Find the roots or vertex, then check signs and endpoints.
Calculate or simplify this relation.
$x^2-20x+m=(x-10)^2+m-100$
Strict positivity requires a strictly positive minimum.

m>100.

Checks and common pitfalls: Strict positivity requires a strictly positive minimum.

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.

13 / Transfer#Your turn

Solve both parts and justify the conditions used. Part A: Does a<b and c<b guarantee a<c? Explain. Part B: Calculate the discriminant and interpret its sign.

\begin{gathered}\text{B: }x^2-14x+50=0\end{gathered}

Use the domain, units and sampling assumptions stated in the question.
Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

A: Track signs and legal operations before transforming an inequality.

Hint 2

B: Find the roots or vertex, then check signs and endpoints.

Worked solution

Part A reasoning
Track signs and legal operations before transforming an inequality.
Calculate or simplify this relation.
$a=13,\ c=12,\ b=14$
Transitivity requires a chain, not merely a shared upper bound.
Part B reasoning
Find the roots or vertex, then check signs and endpoints.
Calculate or simplify this relation.
$\Delta=(-14)^2-4(50)=-4$
A negative discriminant means no real root.

A: No; the two values below b may occur in either order. B: The requested value is -4.

Checks and common pitfalls: Transitivity requires a chain, not merely a shared upper bound. A negative discriminant means no real root.

Reasoning checklist · self / teacher assessment

Solve part A with its stated restrictions.
Solve part B using an appropriate representation.
Give the reasoning and check conditions in both parts.

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

14 / Transfer#Your turn

Minimize the sum of two positive numbers with this product.

ab=144

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Check nonnegativity, apply AM–GM and verify attainable equality.

Hint 2

With fixed product, AM–GM gives a lower sum bound.

Worked solution

Check nonnegativity, apply AM–GM and verify attainable equality.
Calculate or simplify this relation.
$a+b\ge2\sqrt{ab}=24;\quad a=b=12$
Give an equality pair to prove attainment.

The requested value is 24.

Checks and common pitfalls: Give an equality pair to prove attainment.

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

15 / Transfer#Your turn

Solve the quadratic inequality.

(x-11)(x-14)\le0

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Find the roots or vertex, then check signs and endpoints.

Hint 2

The upward-opening quadratic is nonpositive between its roots.

Worked solution

Find the roots or vertex, then check signs and endpoints.
Calculate or simplify this relation.
$11\le x\le14$
The non-strict comparison includes both roots.

x∈[11,14].

Checks and common pitfalls: The non-strict comparison includes both roots.

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.

End-of-lesson check and correction

Review your latest checked answers and explanations. A draft change requires a fresh check. Written work needs your self-assessment or a teacher’s review.

Enable JavaScript for a summary of your local work.

Choose a foundation skill to revisit ↗

Teacher preparation and assessment

Question sequence

Ask students to name the relevant condition before calculating.

Board plan

Compare valid methods and annotate their conditions.

Anticipated thinking

A correct final value may still hide a missing assumption.

Assessment checklist

Check the method, conditions, reasoning and interpretation separately.