The six cases of the parabola

Combining the sign of $a$ ($a>0$ vs $a<0$) with the sign of $\Delta$ ($\Delta>0$, $\Delta=0$, $\Delta<0$) gives six cases, summarised in the figure below. Learn them by heart: every second-degree inequality falls into one of these six patterns.

The six cases obtained by combining the sign of $a$ (concavity) with the sign of $\Delta$ (number of zeros).

In brief — The six practical rules

$a$	$\Delta$	$y > 0$ when	$y < 0$ when
$>0$	$>0$	$x<x_1$ or $x>x_2$	$x_1<x<x_2$
$>0$	$=0$	$\forall x \neq x_0$	never
$>0$	$<0$	$\forall x\in\mathbb{R}$	never
$<0$	$>0$	$x_1<x<x_2$	$x<x_1$ or $x>x_2$
$<0$	$=0$	never	$\forall x\neq x_0$
$<0$	$<0$	never	$\forall x\in\mathbb{R}$

Watch out — The case $\Delta=0$

When $\Delta = 0$ the statement ”$y\ge 0$ everywhere” or ”$y\le 0$ everywhere” remains true, but be careful about the case of equality: at the point of tangency $x_0$ the value is exactly $0$, not strictly positive or negative. So:

• $a>0,\, \Delta=0$: $y\ge 0$ always; $y > 0$ for every $x\neq x_0$; $y \le 0$ only at $x=x_0$.
• $a<0,\, \Delta=0$: $y\le 0$ always; $y < 0$ for every $x\neq x_0$; $y \ge 0$ only at $x=x_0$.

Links

Topics: Inequalities
Concepts: Concavity · Discriminant · Second-degree inequality · Parabola
Skills: Interpreting a graph · Reasoning by cases