A self-consistent analytical solution for binodal concentrations of the two-component Flory-Huggins model is derived. We show that this form extends the validity of the Ginzburg-Landau expansion away from the critical point to cover the whole phase space. Furthermore, this analytical solution reveals an exponential scaling law of the dilute phase binodal concentration as a function of interaction strength and chain length. We demonstrate explicitly the power of this approach by fitting experimental protein liquid-liquid phase separation boundaries to determine the effective chain length and the solute-solvent interaction energy. Moreover, we demonstrate that this strategy allows us to resolve the differences in the interaction energy contributions of individual amino acids. This analytical framework can serve as a new way to decode the protein sequence grammar for liquid-liquid phase separation.