g10-03-02 · Core topics
How many real zeros?
Can a quadratic be positive without crossing the axis?
1 MIN · PREDICT ON PAPER
Change c in ax²+bx+c; predict two, one or no real zeros before revealing Δ.
Discuss or write first. No typed answer is required to continue.
Quadratic: two distinct real roots
2 MIN · EXPLAIN AND REVISE
Which condition does your claim need?
Compare at least two cases. Write a reason, then check the explanation together.
Reveal explanation and conditions
The discriminant counts real zeros for a≠0. If a=0, the expression is linear or constant, so use its own rule. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.
Conditions: a,b,c are real numbers; quadratic classification requires a≠0.
Mathematical justification: For a≠0, ax²+bx+c=a[(x+b/(2a))²−Δ/(4a²)]. A square equals Δ/(4a²) at zeros, yielding two, one or zero real roots according to Δ.
1 MIN · EXIT CHECK
How many real zeros does x²+1 have? What sign does it have?
Check after answering
No real zero; it is at least 1 and therefore strictly positive.
Teacher guidance and extension
Before: Quadratic equations and squares
Ask: Ask for a numerical invariant deciding the root count beyond the plotted samples.
Watch for: A graph outside the current viewing window is not evidence that a root is absent.
Challenge: Use completing the square to explain why a repeated root does not reverse sign.
Suggested timing: predict 1 min, compare 3 min, explain 2 min, exit check 1 min. Observations can suggest a claim; they do not prove it for every input.
Optional local notebook
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