Chapter 6.5, Problem 54E

To determine

To graph:

Given inequality by related system of equations.

Explanation of Solution

Given information:

  $-(\mathrm{x}+3)(\mathrm{x}-2)={\mathrm{x}}^2-6\mathrm{x}$

Graph:

  

Interpretation:

The graph of the given inequality is,

  $-(\mathrm{x}+3)(\mathrm{x}-2)={\mathrm{x}}^2-6\mathrm{x}$

Now add $6\mathrm{x}$ to both sides

  ${\mathrm{x}}^2-\mathrm{x}+6+6\mathrm{x}={\mathrm{x}}^2-6\mathrm{x}+6\mathrm{x}$

Simplify and subtract ${\mathrm{x}}^2$ from both sides

  $\begin{array}{l}-{\mathrm{x}}^2+5\mathrm{x}+6={\mathrm{x}}^2\\ -{\mathrm{x}}^2+5\mathrm{x}+6-{\mathrm{x}}^2={\mathrm{x}}^2-{\mathrm{x}}^2\\ -2{\mathrm{x}}^2+5\mathrm{x}+6=0\end{array}$

Here, $\mathrm{a}=-2,\mathrm{b}=5,\mathrm{c}=6$ for a quadratic equation of the form $\mathrm{a}{\mathrm{x}}^2+\mathrm{b}\mathrm{x}+\mathrm{c}=0$ the solutions are

  $\begin{array}{l}{\mathrm{x}}_1,2=\frac{-\mathrm{b}\pm \sqrt{{\mathrm{b}}^2-4\mathrm{a}\mathrm{c}}}{2\mathrm{a}}\\ \mathrm{x}=\frac{-5-\sqrt{5^2-4\left(-2\right)6}}{2\left(-2\right)}\\ \mathrm{x}=\frac{-5-\sqrt{73}}{4}\\ \mathrm{x}=\frac{5+\sqrt{73}}{4}\end{array}$

Thus, solution for given equation is $\mathrm{x}=\frac{5+\sqrt{73}}{4}$ .