Why would you think you need to use fsolve? Even without the use of fimplicit this is just a hyperbola.
f = @(x,y) y.*(1 + y/2) - x.^2;
You want something of an easier method than this?
Sigh. This was probably homework. But then, the use of fimplicit is probably not allowed in homework. In that case, suppose you knew the value of x. Could you then solve for y? Hint: how would the quadratic forrmula apply here? Could you find both roots of the quadratic in y, as a function of x?