Chapter 3.5, Problem 43AYU

Explain the circumstances under which the $x\text{-intercepts}$ of the graph of a quadratic function are included in the solution set of a quadratic inequality.

To determine

To find: To explain the circumstances under which the $x\text{-intercepts}$ of the graph of a quadratic function are included in the solution set of a quadratic inequality.

Answer to Problem 43AYU

Solution:

The $x\text{-intercepts}$ of this quadratic function are $x=2$ and $x=3$; which are included in the solution set of the corresponding quadratic inequality.

Explanation of Solution

Let the quadratic function be $f\left(x\right)=x^2-5x+6$

Let the corresponding quadratic inequality be:

$x^2-5x+6\le 0$

Then we get,

$x^2-5x+6\le 0$

$\left(x-2\right)\left(x-3\right)\le 0$

$2\le x\le 3$

The $x\text{-intercepts}$ of this quadratic function are $x=2$ and $x=3$; which are included in the solution set of the corresponding quadratic inequality.