Answer
$x=\dfrac{2}{5}$
Work Step by Step
Squaring both sides of the given equation, $
x=\sqrt{\dfrac{6-13x}{5}}
,$ results to
\begin{align*}\require{cancel}
(x)^2&=\left(\sqrt{\dfrac{6-13x}{5}}\right)^2
\\\\
x^2&=\dfrac{6-13x}{5}
.\end{align*}
Using the properties of equality, the equation above is equivalent to
\begin{align*}
5\cdot x^2&=\dfrac{6-13x}{\cancel5}\cdot\cancel5
\\\\
5x^2&=6-13x
\\\\
5x^2+13x-6&=0
.\end{align*}
Using factoring of trinomials, the equation above is equivalent to
\begin{align*}
(5x-2)(x+3)&=0
.\end{align*}
Equating each factor to zero (Zero Product Property) and solving for the variable, then
\begin{array}{l|r}
5x-2=0 & x+3=0
\\
5x=2 & x=-3
\\\\
x=\dfrac{2}{5}
.\end{array}
Checking the solutions by substitution in the given equation results to
\begin{array}{l|r}
\text{If }x=\dfrac{2}{5}: & \text{If }x=-3
\\\\
\dfrac{2}{5}\overset{?}=\sqrt{\dfrac{6-13\left(\frac{2}{5}\right)}{5}} &
-3\overset{?}=\sqrt{\dfrac{6-13(-3)}{5}}
\\\\
\dfrac{2}{5}\overset{?}=\sqrt{\dfrac{6-\frac{26}{5}}{5}} &
-3\ne\text{some nonnegative number}
\\\\
\dfrac{2}{5}\overset{?}=\sqrt{\dfrac{\frac{30}{5}-\frac{26}{5}}{5}}
\\\\
\dfrac{2}{5}\overset{?}=\sqrt{\dfrac{\frac{4}{5}}{5}}
\\\\
\dfrac{2}{5}\overset{?}=\sqrt{\dfrac{4}{25}}
\\\\
\dfrac{2}{5}\overset{\checkmark}=\dfrac{2}{5}
.\end{array}
Since $x=-3$ does not satisfy the original equation, then the only solution of the equation $
x=\sqrt{\dfrac{6-13x}{5}}
$ is $x=\dfrac{2}{5}$.
