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. 

A Mathematica code for $\displaystyle{x^{2} - \,\sqrt{\, x\, }\, - 2 = 0}$ which changes its sign in $\displaystyle{\left[1,2\right]}$ such that the 'starting guess' is $\displaystyle{{1 + 2 \over 2} = {3 \over 2}}$.

(* x^2 - Sqrt[x] - 2 = 0 *)

dRoot[x_] :=
  Module[{den, num, sqrtX},
   sqrtX = Sqrt[x];
   num = (2 - x^2) sqrtX + x;
   den = 4 x sqrtX - 1;
   2 num/den];

myTol = 1.05 Module[{mp, x = 1.0},
    Do[mp = x;
       x /= 2.0; 
       If[(1.0 + x) > 1.0, Continue[], Break[]], {100000}];
       Sqrt[mp]];

(* Newton-Rapson *)
 Module[{dr, r = 3/2, rAnt, tol},
  Do[rAnt = r;
     dr = dRoot[r];
     r += dr;
     tol = Abs[rAnt + dr/2];
     tol = If[tol > 0, Abs[dr]/tol, Abs[dr]];
     If[tol < myTol, Break[], Continue[]], {100000}
   ];
 N[r, 50]]

Root $\displaystyle{\approx \texttt{1.8311772072083418533865654461108296451696466030113}}$

$\displaystyle{\texttt{myTol} = 1.77017 \times 10^{-7}}$.


It agrees with $\texttt{@Claude Leibovici}$ answer.

