Quadratic inequalities

A quadratic inequality is solved by looking at where the graph of a quadratic function lies above or below the $x$ -axis¹. Let us solve $x^2 - x - 6 > 0$ .

Finding the boundaries

At equality, $x^2 - x - 6 = 0$ gives $(x - 3)(x + 2) = 0$ , so $x = 3$ and $x = -2$ . These are where the parabola meets the $x$ -axis, and they split the number line into three ranges.

Inequality	Where the graph is	Solution
$x^2 - x - 6 > 0$	above the $x$ -axis	$x < -2$ or $x > 3$
$x^2 - x - 6 < 0$	below the $x$ -axis	$-2 < x < 3$

Since this parabola opens upward, it is above the axis outside the two intersection points and below between them.

From the factors alone

The same conclusion follows with no picture. The product $(x - 3)(x + 2)$ is positive exactly when the two factors share a sign: both positive gives $x > 3$ , both negative gives $x < -2$ . The product is negative when the signs disagree, which happens between the roots. Graph or factors, the answer is the same.

A negative leading coefficient

A downward parabola swaps above and below, so an inequality like $-x^2 + x + 6 > 0$ is best handled by multiplying through by $-1$ to get $x^2 - x - 6 < 0$ , remembering that multiplying by a negative number reverses the inequality sign.

When the discriminant is not positive

Case	$> 0$ holds for	$< 0$ holds for
$D = 0$ , opening upward	every $x$ except one point	no $x$
$D < 0$ , opening upward	every real number	no $x$

With no crossings, the answers turn extreme.

An application

Suppose an object thrown straight up has height $h = -5t^2 + 20t$ , with $h$ in meters and $t$ in seconds, and you want the stretch of time during which it is at least $15$ meters high.

\begin{align*} -5t^2 + 20t &\geq 15 \\ t^2 - 4t + 3 &\leq 0 \\ (t - 1)(t - 3) &\leq 0 \end{align*}

So $1 \leq t \leq 3$ . From one second after the throw until three seconds after it, the object is $15$ meters up or higher. Note that dividing by $-5$ reversed the inequality.

For an upward-opening parabola you can remember: greater than $0$ means outside the two roots, less than $0$ means between them. The large dots on the graph are the boundaries.

Quadratic equation, Wikipedia