#### Answer

$x=-4$

#### Work Step by Step

Using the properties of equality, the given equation, $
\sqrt{-x}-6=x
,$ is equivalent to
\begin{array}{l}\require{cancel}
\sqrt{-x}=x+6
.\end{array}
Squaring both sides of the equation and then using the properties of equality, we obtain:
\begin{array}{l}\require{cancel}\left(
\sqrt{-x}
\right)^2=\left(
x+6
\right)^2
\\\\
-x=(x)^2+2(x)(6)+(6)^2
\\\\
-x=x^2+12x+36
\\\\
0=x^2+(12x+x)+36
\\\\
x^2+13x+36=0
\\\\
(x+9)(x+4)=0
\\\\
x=\{-9,-4\}
.\end{array}
Upon checking, only $
x=-4
$ satisfies the original equation.