Chapter 12.1, Problem 24WE

To determine

To solve: The equation $6{\left(z+5\right)}^2=42$ and express irrational solution in simplest radical form.

Answer to Problem 24WE

The solution set is $\left(\sqrt{7}-5,-\sqrt{7}-3\right)$ .

Explanation of Solution

Given information:

The given equation $6{\left(z+5\right)}^2=42$ .

Calculation:

  $\begin{array}{l}\Rightarrow 6{\left(z+5\right)}^2=42\\ \Rightarrow \frac{6{\left(z+5\right)}^2}{6}=\frac{42}{6}\end{array}$

(Divide by 6 on both sides)

  $\begin{array}{l}\Rightarrow {\left(z+5\right)}^2=7\\ \Rightarrow z+5=\pm \sqrt{7}\\ \Rightarrow z+5-5=\pm \sqrt{7}-5\end{array}$

(Subtract 5 from both sides)

  $\begin{array}{l}\Rightarrow z=\pm \sqrt{7}-5\\ \Rightarrow z=\sqrt{7}-5,z=-\sqrt{7}-5\end{array}$

Therefore, the solution set is $\left(\sqrt{7}-5,-\sqrt{7}-3\right)$ .