The general form of quadric surfaces is
$$Ax^2 + By^2 + Cz^2 + Dxy + Eyz + Fxz + Gx + Hy + Iz + J = 0$$
I want to classify all of the possibilities including degenerate cases with the help of general form . I searched on the internet but got really confused ! In the case of conic sections this website gives a complete procedure for classifying : https://brilliant.org/wiki/conics-discriminant/