Answer
$x= 2 $
Work Step by Step
Given \begin{equation}
\sqrt{w+2}= 4-w.
\end{equation} First square both sides of the radical equation to eliminate the radical sign. Rearrange and solve for $w$.
\begin{equation}
\begin{aligned}
\sqrt{w+2}&=4-w\\
\left( \sqrt{w+2}\right)^2&= \left( 4-w\right)^2\\
w+2& =16-8w+w^2\\
0& = 14-9w+w^2.
\end{aligned}
\end{equation} Use the quadratic formula to find the value(s) of $w$.
\begin{equation}
\begin{aligned} a &= 1\ , b= -9\ ,\ c = 14\\
w & = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\\
w&=\frac{-(-9) \pm \sqrt{(-9)^2-4 \cdot 1 \cdot(14)}}{2 \cdot 1} \\
&=\frac{9\pm 5}{2}\\
\implies w& = \frac{9+5}{2}\\
& = 7\\
w& = \frac{9-5}{2}\\
& = 2.
\end{aligned}
\end{equation} Check. \begin{equation}
\begin{aligned}
\sqrt{7+2}& \stackrel{?}{=} 4-7 \\
3& =-3\quad \textbf{False}\\
\sqrt{2+2}& \stackrel{?}{=} 4-2\\
2& =2\quad \checkmark.
\end{aligned}
\end{equation} The solution is $x= 2 $.
