Answer
$\frac{8-\sqrt{3}}{2}$; $\frac{8+\sqrt{3}}{2} $
Work Step by Step
Given \begin{equation}
\frac{2}{7}(p-4)^2-\frac{3}{14}=0.
\end{equation} Apply the square root property to solve: \begin{equation}
\begin{aligned}
\frac{2}{7}(p-4)^2-\frac{3}{14} & =0 \\
\frac{2}{7}(p-4)^2 \cdot \frac{7}{2} & =\frac{3}{14} \cdot \frac{7}{2} \\
(p-4)^2 & =\frac{3}{4} \\
p-4 & = \pm \frac{\sqrt{3}}{2} \\
p & =4 \pm \frac{\sqrt{3}}{2}.
\end{aligned}
\end{equation} This gives: \begin{equation}
\begin{aligned}
& p=4-\frac{\sqrt{3}}{2}=\frac{8-\sqrt{3}}{2} \\
& p=4-\frac{\sqrt{3}}{2}=\frac{8+\sqrt{3}}{2}.
\end{aligned}
\end{equation} Check: \begin{equation}
\begin{aligned}
\frac{2}{7}\left(\frac{8-\sqrt{3}}{2}-4\right)^2-\frac{3}{14} & \stackrel{?}{=}0 \\
\frac{2}{7}\cdot\frac{3}{4}-\frac{3}{14} & \stackrel{?}{=}0 \\
0 & =0\checkmark\\
\frac{2}{7}\left(\frac{8+\sqrt{3}}{2}-4\right)^2-\frac{3}{14} & \stackrel{?}{=}0 \\
\frac{2}{7}\cdot\frac{3}{4}-\frac{3}{14} & \stackrel{?}{=}0 \\
0 & =0\checkmark.
\end{aligned}
\end{equation} The solution is:$$p=\frac{8-\sqrt{3}}{2},\quad p=\frac{8+\sqrt{3}}{2}.$$
