College Algebra, Chapter 2, 2.3, Section 2.3, Problem 4

Below are the graphs of $y = 5x - x^2$ and $y =4$.




a.) Find the solutions of the equation $5x - x^2 = 4$
b.) Find the solutions of the inequality $5x - x^2 > 4$

a.)

$\begin{equation} \begin{aligned} 5x - x^2 &=4 && \text{Model}\\ \\ -5x + x^2 &= -4 && \text{Divide both sides by } -1 \\ \\ x^2 - 5x + \frac{25}{4} &= -4 + \frac{25}{4} && \text{Complete the square: Add } \left( \frac{-5}{2} \right)^2 = \frac{25}{4}\\ \\ \left( x - \frac{5}{2} \right)^2 &= \frac{9}{4} && \text{Perfect square}\\ \\ x &= \pm \sqrt{\frac{9}{4}} && \text{Take the square root}\\ \\ x &= \frac{5}{2} \pm \frac{3}{2} && \text{Add } \frac{5}{2} \end{aligned} \end{equation}$



$\begin{equation} \begin{aligned} x &= \frac{5}{2} + \frac{3}{2} &&\text{and}& x &= \frac{5}{2} - \frac{3}{2} && \text{Solve for } x\\ \\ x &=4 &&\text{and}& x &=1 && \text{Simplify} \end{aligned} \end{equation}$


b.)

$\begin{equation} \begin{aligned} 5x - x^2 &> 4 && \text{Model}\\ \\ -5x + x^2 &< -4 && \text{Divide both sides by } -1 \\ \\ x^2 - 5x + \frac{25}{4} &< -4 + \frac{25}{4} && \text{Complete the square: Add } \left( \frac{-5}{2} \right)^2 = \frac{25}{4}\\ \\ \left( x - \frac{5}{2} \right)^2 &< \frac{9}{4} && \text{Perfect square} \end{aligned} \end{equation}$



$\begin{equation} \begin{aligned} x - \frac{5}{2} &< \sqrt{\frac{9}{4}} &&\text{and}& x - \frac{5}{2} &> - \sqrt{\frac{9}{4}} && \text{Take the square root}\\ \\ x &< \frac{5}{2} + \sqrt{\frac{9}{4}} &&\text{and}& x &> \frac{5}{2} - \sqrt{\frac{9}{4}} && \text{Add } \frac{5}{2}\\ \\ x &< \frac{5}{2} + \frac{3}{2} &&\text{and}& x &> \frac{5}{2} + \frac{3}{2} && \text{Solve for } x\\ \\ x &< 4 &&\text{and}& x &> 1 && \text{Simplify} \end{aligned} \end{equation}$


The solution is $1 < x < 4$