Answer
$x=8$
Work Step by Step
Squaring both sides, the given equation is equivalent to
\begin{array}{l}\require{cancel}
\sqrt{2x}=\sqrt{x+1}+1
\\\\
\left(\sqrt{2x}\right)^2=\left(\sqrt{x+1}+1\right)^2
\\\\
2x=(\sqrt{x+1})^2+2(\sqrt{x+1})(1)+(1)^2
\\\\
2x=x+1+2\sqrt{x+1}+1
\\\\
2x-x-1-1=2\sqrt{x+1}
\\\\
x-2=2\sqrt{x+1}
.\end{array}
Squaring both sides again, the equation above is equivalent to
\begin{array}{l}\require{cancel}
x-2=2\sqrt{x+1}
\\\\
(x-2)^2=(2\sqrt{x+1})^2
\\\\
(x)^2-2(x)(2)+(2)^2=4(x+1)
\\\\
x^2-4x+4=4x+4
\\\\
x^2-4x-4x+4-4=0
\\\\
x^2-8x=0
.\end{array}
Factoring the $GCF=x,$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
x^2-8x=0
\\\\
x(x-8)=0
.\end{array}
Equating each factor to zero and then solving the variable, then the solutions are $
x=\{0,8\}
.$
If $x=0,$ then
\begin{array}{l}\require{cancel}
\sqrt{2x}=\sqrt{x+1}+1
\\\\
\sqrt{2(0)}=\sqrt{0+1}+1
\\\\
\sqrt{0}=\sqrt{1}+1
\\\\
0=1+1
\\\\
0=2
\text{ (FALSE)}
.\end{array}
Hence, only $
x=8
$ satisfies the original equation.
