Based on that, you can plot the answer. I put a Quiet[] in there because sometimes the solution has issues but you can get the idea by doing something like this:

Manipulate[
 ListLinePlot[
  Quiet[ans = 
    Table[x /. 
      FindRoot[
       a^2*Exp[x/a] + x^2/2*(1 - b) - a^2 - a*x*Exp[x/a] == 
        0, {x, -1, -100, -0.001}], {b, 0.01, 0.9, 0.01}]]], {{a, 0.3},
   0.1, 1}]

Giving a plot that looks like this for any value of a (only the y axis seems to change with a). This suggests you can easily do an excellent interpolation or an approximate curve fit.

Thank you for the quick reply. As I expect based on the physical reasoning that stays behind the derivation of this equation and looking at its plot, the solutions of this equation should be:

• x1=x2=0 when b>=1

• x1=0 and x2<0 when 0<b<1

• x1=0 and x2>0 when b<0

So, one solution is always x=0. I'm mainly interested to derive the solution when 0<b<1. For example, when I solve it with NSolve and assuming a = 0.3 and b=0.5, the solutions are x=0 and x=-0.327474 as confirmed by the plot. I guess that the cursory numerical exploration suggested above did not provide negative solutions because the table command did not explore values 0<b<1.