Chapter 9.4, Problem 52HP

To determine

Compare and contrast the following strategies for solving $x^2-5x-7=0$ completing the square, graphing and factoring.

Answer to Problem 52HP

Explanation of Solution

Given:

The equation: $x^2-5x-7=0$

Concept Used:

The leading coefficient is 1, solving the equation by completing the square is little bit simpler.

Calculation:

Solving the equation $x^2-5x-7=0$ by completing the square

  $\begin{array}{l}{x}^2-5x-7=0[\text{original}\text{equation}]\\ $ x^2-5x+\frac{25}{4}-7=0+\frac{25}{4}[\text{Since}{(\frac{5}{2})}^2=1\text{add}\text{to}\text{each}\text{side}]$x^2-5x+\frac{25}{4}-7+7=\frac{25}{4}+7[\text{add}\text{7}\text{to}\text{each}\text{side}]\\ $ {(x-\frac{5}{2})}^2=\frac{53}{4}$\sqrt{{(x-\frac{5}{2})}^2}=\pm \sqrt{\frac{53}{4}}[\text{Take}\text{the}\text{square}\text{root}\text{of}\text{each}\text{side}]\\ x-\frac{5}{2}=\pm \frac{\sqrt{53}}{2}\\ x-\frac{5}{2}+\frac{5}{2}=\frac{5}{2}\pm \frac{\sqrt{53}}{2}[\text{add}\frac{5}{2}\text{to}\text{both}\text{sides}]\\ x=\frac{5}{2}-\frac{\sqrt{53}}{2}\text{or}x=\frac{5}{2}+\frac{\sqrt{53}}{2}[\text{Seperate}\text{the}\text{solution}]\\ x=-1.14\text{or}x=6.14\end{array}$

To solve the equation by graphing, use a graphing calculator.

  

The roots are given as decimal approximations. So, for this equation the solutions would have to be given as estimations.

There are no factors of (− 7) that have a sum of− 5, so solving by factoring is not possible.

Thus,