LEARN · EXPLAIN · REVISE

Classify the real solutions for every real m.

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

TOPIC 01

Quadratic equations and functions

The condition a≠0 is essential; using the quadratic formula at m=0 would divide by zero.

PREDICT → MOVE → EXPLAIN

Before moving a slider, predict the opening, vertex and intercepts. Then compare the graph with your prediction.

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

01 / Standard#Your turn

mx^2+3x-6=0

Working and explanation

BUILD THE REASONING

Hint 1

First separate m=0 from quadratic cases.

Hint 2

For m≠0, use Δ=9+24m.

Worked solution

The zero parameter gives a linear equation.
$m=0\Rightarrow x=2$
Classify the quadratic using the discriminant.
$m\ne0:\quad \Delta=9+24m$
Keep the linear exception distinct.
$m<-3/8:0;\quad m=-3/8:1;\quad m>-3/8,\ m\ne0:2$

m=0 has one linear root; m=−3/8 a repeated root x=4; the other counts follow Δ.

Checks and common pitfalls: The condition a≠0 is essential; using the quadratic formula at m=0 would divide by zero.

Reasoning checklist · self / teacher assessment

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