Answer
The solution is $x=5$
Work Step by Step
$x-\sqrt{x-1}=3\hspace{0.7cm}{\color{blue}{\text{Given equation}}}$
$\Rightarrow x=3+\sqrt{x-1}$
$\Rightarrow x-3=\sqrt{x-1}$
$\Rightarrow (x-3)^2=(\sqrt{x-1})^2$
$\Rightarrow x^2-6x+9=x-1$
$\Rightarrow x^2-6x+9-x+1=0$
$\Rightarrow x^2-7x+10=0$
$\Rightarrow (x-2)(x-5)=0$
$x-2=0\hspace{0.3cm}\text{or}\hspace{0.3cm}x-5=0$
$x=2\hspace{0.3cm}\text{or}\hspace{0.3cm}x=5$
$\underline{\textbf{Check the answer}}$
At $x=2$
LHS $2-\sqrt{2-1}=2-\sqrt{1}=2-1=0\ne$RHS $\hspace{0.3cm}{\color{red}{Reject}}$
At $x=5$
LHS $5-\sqrt{5-1}=5-\sqrt{4}=5-2=3=$RHS
The solution is $x=5$
