Answer
$x=\{1,4\}$
Work Step by Step
Squaring both sides of the given equation, $
\sqrt{9x}=x+2
,$ then,
\begin{array}{l}\require{cancel}
\left( \sqrt{9x} \right)^2=\left( x+2 \right)^2
\\\\
9x=(x)^2+2(x)(2)+(2)^2
\\\\
9x=x^2+4x+4
\\\\
-x^2+9x-4x-4=0
\\\\
-x^2+5x-4=0
.\end{array}
Using $\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$ (or the Quadratic Formula), the solutions of the quadratic equation, $
-x^2+5x-4=0
,$ are
\begin{array}{l}\require{cancel}
\dfrac{-(5)\pm\sqrt{(5)^2-4(-1)(-4)}}{2(-1)}
\\\\=
\dfrac{-5\pm\sqrt{25-16}}{-2}
\\\\=
\dfrac{-5\pm\sqrt{9}}{-2}
\\\\=
\dfrac{-5\pm\sqrt{(3)^2}}{-2}
\\\\=
\dfrac{-5\pm3}{-2}
\\\\=
\dfrac{-5-3}{-2}
\text{ OR }
\dfrac{-5+3}{-2}
\\\\=
\dfrac{-8}{-2}
\text{ OR }
\dfrac{-2}{-2}
\\\\=
4
\text{ OR }
1
.\end{array}
Upon checking, both solutions satisfy the original equation. Hence, $
x=\{1,4\}
.$
