Hi Dia,
Your problem is somewhat unclear. Nevertheless, I hope to understand your requests as close as possible. Firstly, we need to define the function
$f[a]$ as:
f[a_] := G (1 - a/4 + \[Alpha] a^2/8 - 5 \[Gamma] a^3/64) - d
Your
$F[A^2]$ is obtained as
In[2]:= f[A^2]
Out[2]= -0.2 + 0.1 (1 - A^2/4 + 0.01425 A^4 - 0.000234375 A^6)
Let us define the parameters:
\[Alpha] = 0.114;
\[Gamma] = 0.003; G = 0.1; d = 0.2;
Now, the graphs may be plot:
Plot[{f[A^2], Sqrt[A], -Sqrt[A]}, {A, 0, 9}, PlotStyle -> {Red, Blue, Blue}]