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:

\[x = 3 \quad \text{and} \quad x = 2\]

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:

\[\{ x: 2 < x < 3\}\]