2. (ROM) For which real numbers $x$ do the following equations hold: (a) $\sqrt{x+\sqrt{2 x-1}}+\sqrt{x+\sqrt{2 x-1}}=\sqrt{2}$, (b) $\sqrt{x+\sqrt{2 x-1}}+\sqrt{x+\sqrt{2 x-1}}=1$, (c) $\sqrt{x+\sqrt{2 x-1}}+\sqrt{x+\sqrt{2 x-1}}=2$?
2. For the square roots to be real we must have $2 x-1 \geq 0 \Rightarrow x \geq 1 / 2$ and $x \geq \sqrt{2 x-1} \Rightarrow x^{2} \geq 2 x-1 \Rightarrow(x-1)^{2} \geq 0$, which always holds. Then we have $\sqrt{x+\sqrt{2 x-1}}+\sqrt{x-\sqrt{2 x-1}}=c \Longleftrightarrow$  $$ c^{2}=2 x+2 \sqrt{x^{2}-\sqrt{2 x-1}^{2}}=2 x+2|x-1|= \begin{cases}2, & 1 / 2 \leq x \leq 1 \\ 4 x-2, & x \geq 1\end{cases} $$  (a) $c^{2}=2$. The equation holds for $1 / 2 \leq x \leq 1$. (b) $c^{2}=1$. The equation has no solution. (c) $c^{2}=4$. The equation holds for $4 x-2=4 \Rightarrow x=3 / 2$.