Answer
$\left\{\dfrac{1}{2},2\right\}$
Work Step by Step
Squaring both sides, the given equation, $
x\sqrt{2}=\sqrt{5x-2}
,$ is equivalent to
\begin{align*}
\left(x\sqrt{2}\right)^2&=\left(\sqrt{5x-2}\right)^2
\\
x^2(2)&=5x-2
\\
2x^2&=5x-2
\\
2x^2-5x+2&=0
.\end{align*}
Using factoring of trinomials, the equation above is equivalent to
\begin{align*}
(x-2)(2x-1)&=0
.\end{align*}
Equating each factor to zero (Zero Product Property) and solving for the variable, then
\begin{array}{l|r}
x-2=0 & 2x-1=0
\\
x=2 & 2x=1
\\\\
& x=\dfrac{1}{2}
.\end{array}
Hence, the solution set of the equation $
x\sqrt{2}=\sqrt{5x-2}
$ is $\left\{\dfrac{1}{2},2\right\}$.
