Generally speaking, it is impossible to determine three real number variables using just one equation. (If the variables are known to be integers, then it will be the subject of indeterminate equation. Please see the book %%HREF%%Indeterminate Equation%%Home/35-books/95-book-indeterminate-equations%%.) Having said the above, one frequently seen possibility to make such an equation solvable is that it can be written as a sum of several squares. The given equation is equivalent to \begin{align} x+y+z - 2\sqrt{x} - 2\sqrt{y-1} + 2\sqrt{z-2} &= 0\\ (x-2\sqrt{x} + 1) + ((y-1)-2\sqrt{y-1} + 1)+((z-2)-2\sqrt{z-2}+1)&=0\\ (\sqrt{x}-1)^2 + (\sqrt{y-1}-1)^2 + (\sqrt{z-2}-1)^2 &=0 \end{align} For this equation to hold, all the three terms must all equal zero. i.e. $$\sqrt{x}-1=\sqrt{y-1}-1=\sqrt{z-2}-1=0\implies (x, y, z)=\boxed{1, 2, 3}$$