Quadratic Inequalities

When we solve quadratic inequalities, we need to take into account the shape of the graph to consider the direction of our solutions.

Solving a quadratic inequality

$$x^2-5x+6<0$$

Calculate the roots to the quadratic.

$$x=\frac{-(-5)\pm \sqrt{(-5{)}^2-4(1)(6)}}{2(1)}$$

$$=\frac{5\pm \sqrt{25-24}}{2}$$

$$=\frac{5\pm 1}{2}$$

The roots are:

$$x=3\text{and}x=2$$

The quadratic is positive (the coefficient of the $x^2$ term is positive). This means that the graph will be underneath the x-axis between the roots.

$$2<x<3$$

In set notation:

$$\{x:2<x<3\}$$