Kenneth ChengMathematical modeling
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For which m is this positive for every real x?

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

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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

For which m is this positive for every real x?

x2−20x+mx^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

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

  2. Calculate or simplify this relation.

    x2−20x+m=(x−10)2+m−100x^2-20x+m=(x-10)^2+m-100

  3. 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.

Focus on one question

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.

No sign-in. Work stays in this browser. Export before clearing browser data. Written reasoning is assessed with a checklist.

Curriculum and source notes ↗