Is there a method for solving equations like $x^{2}-\sqrt{x}-2=0$? Is there a method for solving equations like $x^{2}-\sqrt{x}-2=0$?
As far as I can remember, I don't know any method for equations like this. 

Setting $y = \sqrt{x}$, the equation becomes
$$
y^4 - y - 2 = 0 .
$$
This is a degree four polynomial equation. In general, it is possible to solve this kind of equations, but it is complicated and beyond the level a precalculus student should master; check the Wikipedia article on Quartic functions if you want to know more.
On a more basic level, it is useful to know that, once you spot an "easy" solution of the equation, finding the remaining solutions becomes easier because the order of the polynomial can be reduced. See the article on Synthetic division for a full explanation. In the case of the equation $y^4 - y -2 = 0$, it is easy to see that $-1$ is a solution. $y = -1$ does not correspond to a solution of your equation since it is negative, but that is not important. Using synthetic division, one obtains
$$
y^4 - y - 2 = (y + 1)(y^3 - y^2 + y - 2)
$$
so to find the other solutions to $y^4 - y -2 = 0$, it suffices to find the solutions of $y^3 - y^2 + y - 2 = 0$. This is slightly easier, since it is easier to find the roots of a cubic polynomial than to find those of a quartic polynomial, although it is still too hard for a precalculus course, I'm afraid.
