LEARN · EXPLAIN · REVISE

For which m is this positive for every real x?

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

TOPIC 01

Quadratic functions, equations and inequalities

Build understanding of quadratic functions, equations and inequalities through definitions, contrasting cases and justified applications.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Standard#Your turn

x^2-6x+m

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

m>9.

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

Reasoning checklist · self / teacher assessment

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

Teacher preparation and assessment

What must be true before using the main rule for quadratic functions, equations and inequalities?
Which representation makes this task easier, and why?
Change one assumption. Does the conclusion survive?

Apply quadratic functions, equations and inequalities with explicit conditions.
Signs and roots: Roots divide the number line into intervals of fixed sign.
Vertex and discriminant: Complete the square for extrema and use the discriminant to count roots.
$\Delta=b^2-4ac$
Close with: conditions → representation → reasoning → check.

Expected reasoning: Roots divide the number line into intervals of fixed sign.
Expected reasoning: Complete the square for extrema and use the discriminant to count roots.
Expected correction: Treat a=0 separately.