Lesson 6: Quadratic Inequalities

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

A quadratic inequality in two variables has the form $y < ax^2 + bx + c$, $y > ax^2 + bx + c$, $y \leq ax^2 + bx + c$, or $y \geq ax^2 + bx + c$, where $a \neq 0$. The solution is a region in the coordinate plane bounded by the parabola $y = ax^2 + bx + c$.

Examples

Property

For quadratic inequalities in two variables:

• Use a dashed line for strict inequalities: $y < ax^2 + bx + c$ or $y > ax^2 + bx + c$
• Use a solid line for non-strict inequalities: $y \leq ax^2 + bx + c$ or $y \geq ax^2 + bx + c$

Examples