Quadratic Inequalities

Solving quadratic inequalities combines the quadratic formula with reasoning about the shape of a parabola.

General Method

⚡Theorem — Method for Solving Quadratic Inequalities

To solve $ax^2 + bx + c > 0$ (or $\geq$ , $<$ , $\leq$ ):

The solution depends on the discriminant $D = b^2 - 4ac$ and the sign of $a$ .

The parabola crosses the $x$ -axis at $r_1$ and $r_2$ .

$a > 0$ : The quadratic is positive outside the roots ( $x < r_1$ or $x > r_2$ ) and negative between them ( $r_1 < x < r_2$ ).
$a < 0$ : The quadratic is negative outside the roots and positive between them.

✎Example — Two Distinct Roots

Solve $x^2 - 5x + 6 > 0$ .

Solution. Factor: $(x - 2)(x - 3) > 0$ . Roots: $r_1 = 2$ , $r_2 = 3$ .

Since $a = 1 > 0$ , the parabola opens up. It is positive outside the roots:

$x < 2 \quad \text{or} \quad x > 3$

In interval notation: $(-\infty, 2) \cup (3, +\infty)$ .

The quadratic equals $a(x - r)^2$ .

$a > 0$ : $(x - r)^2 \geq 0$ always. The quadratic is $\geq 0$ for all $x$ , equal to $0$ only at $x = r$ . Strictly positive for $x \neq r$ .
$a < 0$ : The quadratic is $\leq 0$ for all $x$ .

✎Example — Repeated Root

Solve $-x^2 + 4x - 4 \leq 0$ .

Solution. Factor: $-(x - 2)^2 \leq 0$ .

Since $-(x-2)^2 \leq 0$ for all real $x$ (a non-positive square times $-1$ ), the inequality holds for all real numbers: $x \in \mathbb{R}$ .

The parabola never crosses the $x$ -axis.

✎Example — No Real Roots

Solve $x^2 + x + 1 > 0$ .

Solution. $D = 1 - 4 = -3 < 0$ , and $a = 1 > 0$ .

The parabola opens up and never crosses the $x$ -axis, so $x^2 + x + 1 > 0$ for all real $x$ . The solution is $\mathbb{R}$ .

✏Exercise

Solve $x^2 - 3x - 10 < 0$ . Find the roots, determine the sign, and write the solution in interval notation.