Answer
$6-4 \sqrt{2}$; $6+4 \sqrt{2} $
Work Step by Step
Given \begin{equation}
-0.25(x-6)^2+8 =0.
\end{equation} Apply the square root property to solve: \begin{equation}
\begin{aligned}
\left(-0.25(x-6)^2+8\right)(-4) & =0(-4) \\
(x-6)^2-32 & =0 \\
(x-6)^2 & =32 \\
(x-6)^2 & =16 \cdot 2 \\
x-6 & = \pm \sqrt{16 \cdot 2} \\
x-6 & = \pm 4 \sqrt{2}.
\end{aligned}
\end{equation} This gives: \begin{equation}
\begin{aligned}
&\begin{aligned}
x & =6-4 \sqrt{2} \\
& \approx 0.343
\end{aligned}\\
&\begin{aligned}
x & =6+4 \sqrt{2} \\
& \approx 11.657
\end{aligned}
\end{aligned}
\end{equation} Check: \begin{equation}
\begin{array}{r}
-0.25(0.343-6)^2+8 \stackrel{?}{=}0 \\
-8+8\stackrel{?}{=}0 \\
0=0\checkmark\\
-0.25(11.657-6)^2+8 \stackrel{?}{=}0 \\
-8+8\stackrel{?}{=}0 \\
0=0\checkmark.
\end{array}
\end{equation} The solution is:$$x=6-4 \sqrt{2},\quad x=6+4 \sqrt{2}.$$
