← Senior Mathematics Studio

LEARN · EXPLAIN · REVISE

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?

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

Return to the lesson / paper ↗

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 / 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.

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.