Abstract: In this paper, we prove that the binary Goppa codes with L=GF(2^m) and G(z)=z²+az+b are quasi-perfect unless m is even and S₁=0,S₃=a^-1. When m is even, there is not any vector of weight <3 having syndrome S₁=0 and S₃=a^-1 but there is at least a vector of weight 4 having syndrome S₁=0 and S₃=a^-1. At the same time we give also a complete decoding of the binary Goppa codes with L=GF(2^m) and G(z)=z²+az+b.