Answer
$m=8$
Work Step by Step
Squaring both sides, the given equation, $
\sqrt{m+1}=-1+\sqrt{2m}
,$ is equivalent to
\begin{align*}
\left(\sqrt{m+1}\right)^2&=\left(-1+\sqrt{2m}\right)^2
\\
m+1&=(-1)+2(-1)(\sqrt{2m})+(\sqrt{2m})^2
&(\text{use }(a+b)^2=a^2+2ab+b^2)
\\
m+1&=1-2\sqrt{2m}+2m
\\
2\sqrt{2m}&=(2m-m)+(1-1)
\\
2\sqrt{2m}&=m
.\end{align*}
Squaring both sides again, the equation above is equivalent to
\begin{align*}
\left(2\sqrt{2m}\right)^2&=(m)^2
\\
4(2m)&=m^2
\\
8m&=m^2
\\
0&=m^2-8m
\\
m^2-8m&=0
.\end{align*}
Factoring the $GCF=m$, the equation above is equivalent to
\begin{align*}
m(m-8)&=0
.\end{align*}
Equating each factor to zero (Zero Product Property) and solving for the variable, then
\begin{array}{l|r}
m=0 & m-8=0
\\
& m=8
.\end{array}
Checking by substituting the solutions in the given equation results to
\begin{array}{l|r}
\text{If }m=0: & \text{If }m=8:
\\\\
\sqrt{0+1}\overset{?}=-1+\sqrt{2(0)} &
\sqrt{8+1}\overset{?}=-1+\sqrt{2(8)}
\\
\sqrt{1}\overset{?}=-1+\sqrt{0} &
\sqrt{9}\overset{?}=-1+\sqrt{16}
\\
\sqrt{1}\overset{?}=-1+0 &
3\overset{?}=-1+4
\\
1\ne-1 &
3\overset{\checkmark}=3
.\end{array}
Since $m=0$ does not satisfy the original equation then the only solution of the equation $
\sqrt{m+1}=-1+\sqrt{2m}
$ is $m=8$.
