In this thesis we are studying the surfaces in the Euclidean 3-space R2 described by the polynomial of second degree of the form: F(x,y,z)= a11x2 +2a12xy +2a13xz +a22y2 +2a23yz +a33z2 +b1x +b2y +b3z +c We give the proof of a classical theorem stating that there exist exactly 9 different surfaces described by this kind of equations: Ellipsoid, Elliptic Paraboloid, Hyperbolic Paraboloid, Elliptic Cylinder, Hyperbolic Cylinder, Parabolic Cylinder, Elliptic Hyperboloid of two Sheets, Elliptic Cone, Elliptic Hyperboloid of one Sheet Keywords Second-degree curves, Second-degree surfaces, Orthogonal diagonalization, ON-matrix