Answer
$x=3$
Work Step by Step
Squaring both sides of the given equation, $
2x=\sqrt{11x+3}
,$ results to
\begin{align*}\require{cancel}
(2x)^2&=\left(\sqrt{11x+3}\right)^2
\\
4x^2&=11x+3
\\
4x^2-11x-3&=0
.\end{align*}
Using factoring of trinomials, the equation above is equivalent to
\begin{align*}
(4x+1)(x-3)&=0
.\end{align*}
Equating each factor to zero (Zero Product Property) and solving for the variable, then
\begin{array}{l|r}
4x+1=0 & x-3=0
\\
4x=-1 & x=3
\\\\
x=-\dfrac{1}{4}
.\end{array}
Checking the solutions by substitution in the given equation results to
\begin{array}{l|r}
\text{If }x=-\dfrac{1}{4}: & \text{If }x=3:
\\\\
2\left(-\dfrac{1}{4}\right)\overset{?}=\sqrt{11\left(-\dfrac{1}{4}\right)+3} &
2(3)\overset{?}=\sqrt{11(3)+3}
\\\\
-\dfrac{1}{2}\overset{?}=\sqrt{-\dfrac{11}{4}+3} &
6\overset{?}=\sqrt{33+3}
\\\\
-\dfrac{1}{2}\overset{?}=\sqrt{-\dfrac{11}{4}+\dfrac{12}{4}} &
6\overset{?}=\sqrt{36}
\\\\
-\dfrac{1}{2}\overset{?}=\sqrt{\dfrac{1}{4}} &
6\overset{\checkmark}=6
\\\\
-\dfrac{1}{2}\ne\dfrac{1}{2}
.\end{array}
Since $x=-\dfrac{1}{4}$ does not satisfy the original equation, then the only solution of the equation $
2x=\sqrt{11x+3}
$ is $x=3$.
