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. 

If you use the substitution $u=\sqrt x$ you see that your equation is basically a quartic. Since we can solve all polynomials of degree less than or equal to $4$, there is a "method" to solve it, but it's not necessarily easy.
As it turns out, there are two real solutions to $u^4-u-2=0$. The first is $-1$, but that turns out to be an extra root introduced when we did the substitution $u=\sqrt x$ (we cannot plug $-1$ back in to the formula or we have a $\sqrt {-1}$ term).
The other root is near $1.3$, If we solve this back for $x$, we get $x=1.3^2$ which is near $1.83$.
Thus, there is one real root to the original equation which is the square of the second root given by Wolfram here.
