(* --------------------------------------------------------------- *)
(*   wind     Winding number solution of z^2 + z - 3/4 = 0        *)
(* ---------------------------------------------------------------- *)
(* Exact solutions are z = 1/2 and z = -3/2  *)
(* Graphs show p(z(t)) and z(t) [RED] *)

wind[a_,b_] := ParametricPlot[{{a^2 Cos[2 t] +a Cos[t]-3/4, a^2 Sin[2 t]+a Sin[t]},
{a Cos[t],a  Sin[t]}},{t,0,2 Pi},
PlotRange->{{-b,b},{-b,b}}]

(* w = 2 *)

wind[1.6,4]

(*  w = 1 *)
wind[0.6,1.3]

(* w = 0 *)

wind[0.4,1.2]