How can I figure out a nested radical?

Question

I have to find the value of: ( The picture )

And I have this solution:

Now, I understood how they took $x = \sqrt{1+2x}$ $\implies x^2-2x-1= 0$

But how did they take $(x-1)^2 = 2$?

Answer 1

$$ x^2-2x-1 = \underbrace{x^2-2x+\color{red}{1}}-\color{red}{1}-1 = (x-1)^2-1-1 = (x-1)^2-2$$

so $$ x^2-2x-1 = 0\implies (x-1)^2=2$$