Lesson 9.8: Solve Quadratic Inequalities

New Concept

We'll explore quadratic inequalities, which are like quadratic equations but use symbols like $>, <, \geq, \leq$. You'll learn to find solutions both visually by graphing parabolas and systematically using algebraic methods with critical points.

What’s next

Next, you'll work through interactive examples to master solving these inequalities graphically and algebraically, followed by practice cards to test your skills.

Property

A quadratic inequality is an inequality that contains a quadratic expression.
The standard form of a quadratic inequality is written:

When we ask when is $ax^2 + bx + c < 0$, we are asking when is $f(x) < 0$. We want to know when the parabola is below the $x$-axis. When we ask when is $ax^2 + bx + c > 0$, we are asking when is $f(x) > 0$. We want to know when the parabola is above the $x$-axis.

Examples

• The expression $x^2 - 4x + 3 > 0$ is a quadratic inequality asking for x-values where the parabola is above the x-axis.

• The expression $-2y^2 + 5y - 2 \leq 0$ is a quadratic inequality asking for y-values where the parabola is on or below the y-axis.

Property

How to Solve a quadratic inequality graphically.

Step 1. Write the quadratic inequality in standard form.

Step 2. Graph the function $f(x) = ax^2 + bx + c$.

Property

How to Solve a quadratic inequality algebraically.

Step 1. Write the quadratic inequality in standard form.

Step 2. Determine the critical points—the solutions to the related quadratic equation, $ax^2 + bx + c = 0$.

Property

When the related quadratic equation $ax^2 + bx + c = 0$ has complex solutions, the discriminant $b^2 - 4ac$ is negative. This means the graph of the parabola does not intercept the $x$-axis.

If the parabola opens upward ($a>0$), it is entirely above the $x$-axis ($ax^2 + bx + c > 0$ for all $x$).

If the parabola opens downward ($a<0$), it is entirely below the $x$-axis ($ax^2 + bx + c < 0$ for all $x$).