Answer
$x=1$
Work Step by Step
Squaring both sides of the given equation, $
3x=\sqrt{8x+1}
,$ then,
\begin{array}{l}\require{cancel}
\left( 3x \right)^2=\left( \sqrt{8x+1} \right)^2
\\\\
9x^2=8x+1
\\\\
9x^2-8x-1=0
.\end{array}
Using $\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$ (or the Quadratic Formula), the solutions of the quadratic equation, $
9x^2-8x-1=0
,$ are
\begin{array}{l}\require{cancel}
\dfrac{-(-8)\pm\sqrt{(-8)^2-4(9)(-1)}}{2(9)}
\\\\=
\dfrac{8\pm\sqrt{64+36}}{18}
\\\\=
\dfrac{8\pm\sqrt{100}}{18}
\\\\=
\dfrac{8\pm\sqrt{(10)^2}}{18}
\\\\=
\dfrac{8\pm10}{18}
\\\\=
\dfrac{8-10}{18}
\text{ OR }
\dfrac{8+10}{18}
\\\\=
\dfrac{-2}{18}
\text{ OR }
\dfrac{18}{18}
\\\\=
\dfrac{-1}{9}
\text{ OR }
1
.\end{array}
Upon checking, only $
x=1
$ satisfies the original equation.