Case a positive, Delta positive

The most common standard case: a parabola with upward concavity and two distinct zeros. The inequality with sense $<$ asks where the parabola lies below the $x$-axis, that is, between the two zeros.

Example — Case $a>0$, $\Delta>0$

Solve $x^2-12x < 0$.

Concavity: $a=1>0$, parabola opening upwards. Zeros: $x(x-12)=0 \implies x_1=0,\,x_2=12$. $\Delta=144>0$.

We look for where the parabola lies below the $x$-axis: between the two zeros.

The parabola lies below the $x$-axis in the interval between the two zeros.

Solution: $\boxed{0 < x < 12}$.

Links

Topics: Inequalities
Concepts: Discriminant · Second-degree inequality · Parabola
Methods: Second-degree inequality · Parabola vertex
Skills: Interpreting a graph · Solving inequalities