Concept preview

What you will be able to do

1. 01Understand a quadratic inequality as asking where a parabola is above or below the $x$-axis.
2. 02Solve $ax^2 + bx + c > 0$ (or $<, \ge, \le$) by finding the roots and reading the parabola's sign.
3. 03Choose open or closed boundaries correctly for strict vs. inclusive inequalities.
4. 04Express solutions in inequality and interval notation.
5. 05Handle special cases: perfect squares, and quadratics with no real roots.

Solving the equation $x^2 - 5x + 6 = 0$ pins down the two isolated spots where the parabola touches the $x$-axis. A quadratic inequality like $x^2 - 5x + 6 < 0$ asks a bigger, more interesting question: for which values of $x$ is the parabola actually below the axis? The answer is no longer a handful of points but whole continuous intervals of the number line. Learning to find those intervals is the goal of this topic, and the good news is that the parabola's own shape does most of the work.