The Blume-Emery-Griffiths (BEG) model is a spin lattice system where a spin value in {-1,0,1} is assigned to each vertex of Z^d and its Hamiltonian depends on two parameters X and Y. While this model has connections with classical Ising and other spin lattice systems, in some regions of the phase diagram (X,Y) it exhibits behaviors which are not yet completely understood. This talk aims to present the progress concerning the point of intersection of the interfaces of the phase diagram, called the ferromagnetic-antiquadrupolar-disordered (FAD) point. For the two-dimensional case at zero temperature, developments by P.C. Lima, R. Mariani, A. Procacci, and B. Scoppola, using a coupling with sub-critical site percolation produced rigorous results regarding the vanishing of magnetization and decay of correlations. Recently, a new approach using a random cluster representation has been considered to derive results for finite temperatures, aiming to prove the absence of phase transition in the two-dimensional system. This is a joint work with B. Scoppola, A. Procacci, and R. Sanchis.