## Elementary Algebra

$x=-4$
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.