Answer
$x=\dfrac{8}{9}$
Work Step by Step
Squaring both sides of the given equation, $
3x=\sqrt{16-10x}
,$ results to
\begin{align*}\require{cancel}
(3x)^2&=\left(\sqrt{16-10x}\right)^2
\\
9x^2&=16-10x
\\
9x^2+10x-16&=0
.\end{align*}
Using factoring of trinomials, the equation above is equivalent to
\begin{align*}
(9x-8)(x+2)&=0
.\end{align*}
Equating each factor to zero (Zero Product Property) and solving for the variable, then
\begin{array}{l|r}
9x-8=0 & x+2=0
\\
9x=8 & x=-2
\\\\
x=\dfrac{8}{9} & x=-2
.\end{array}
Checking the solutions by substitution in the given equation results to
\begin{array}{l|r}
\text{If }x=\dfrac{8}{9}: & \text{If }x=-2:
\\\\
3\left(\dfrac{8}{9}\right)\overset{?}=\sqrt{16-10\left(\dfrac{8}{9}\right)} &
3(-2)\overset{?}=\sqrt{16-10(-2)}
\\\\
\cancelto13\left(\dfrac{8}{\cancelto39}\right)\overset{?}=\sqrt{16-\dfrac{80}{9}} &
-6\overset{?}=\sqrt{16+20}
\\\\
\dfrac{8}{3}\overset{?}=\sqrt{\dfrac{144}{9}-\dfrac{80}{9}} &
-6\overset{?}=\sqrt{36}
\\\\
\dfrac{8}{3}\overset{?}=\sqrt{\dfrac{64}{9}} &
-6\ne6
\\\\
\dfrac{8}{3}\overset{\checkmark}=\dfrac{8}{3}
.\end{array}
Since $x=-2$ does not satisfy the original equation, then the only solution of the equation $
3x=\sqrt{16-10x}
$ is $x=\dfrac{8}{9}$.
