Answer
$-6-\sqrt{5}$; $-6+\sqrt{5} $
Work Step by Step
Given \begin{equation}
\frac{1}{2}(c+6)^2-\frac{5}{2} =0.
\end{equation} Apply the square root property to solve: \begin{equation}
\begin{aligned}
\frac{1}{2}(c+6)^2-\frac{5}{2} & =0 \\
\left(\frac{1}{2}(c+6)^2\right) 2 & =\left(\frac{5}{2}\right) \cdot 2 \\
(c+6)^2 & =5 \\
c+6 & = \pm \sqrt{5} \\
c & =-6 \pm \sqrt{5}.
\end{aligned}
\end{equation} This gives: \begin{equation}
\begin{aligned}
& c=-6-\sqrt{5} \approx-8.236 \\
& c=-6+\sqrt{5} \approx-3.764
\end{aligned}
\end{equation} Check: \begin{equation}
\begin{aligned}
\frac{1}{2}(-6-\sqrt 5+6)^2-\frac{5}{2} &\stackrel{?}{=}0 \\
\frac{1}{2}(5)-\frac{5}{2}&\stackrel{?}{=}0\\
0 & =0\checkmark\\
\frac{1}{2}(-6+\sqrt 5+6)^2-\frac{5}{2} &\stackrel{?}{=}0 \\
\frac{1}{2}(5)-\frac{5}{2}&\stackrel{?}{=}0\\
0 & =0\checkmark.
\end{aligned}
\end{equation} The solution is:$$x=-6-\sqrt{5},\quad x=-6+\sqrt{5}.$$
