If F is a quadratic function on a n-dimensional vector space,
then F(x) is affine equivalent to one of the following:

  Sum( epsilon[j]*x[j]^2, j=1 ..r  ), 
  Sum( epsilon[j]*x[j]^2, j=1 ..r  ) + alpha, 
  Sum( epsilon[j]*x[j]^2, j=1 ..r-1) + x[r]

where epsilon = +-1 and r <= n, alpha a real non-zero constant.

If one only wants the quadratic hypersurface { x | F(x) = 0 }, then in
the second case one can achieve alpha = 1.

Affine equivalent means, that there is some invertible matrix C and some
translation vector v, such that G(x) = F( x.C + v ). So that is a linear
and invertible change of coordinates (up to translation).

The attached sheet shows, how to compute that representation explicitely
by using examples (being too lame for a proper module).

I did that stuff, since posting a question and missing answers indicate,
that no short and usable sheet is available from people reading that forum.

www.mapleprimes.com/files/102_affineEquivalence_quadraticFunction_MP1.mws

Edited: I corrected the above statement and uploaded the according sheet
(formerly I asserted alpha = 1 can be taken for the function).