In one, two, and three variables, functions in quadratic form are:
F(x) = ax2
F(x, y) = ax2 + by2 + cxy
F(x, y, z) = ax2 + by2 + cz2 + dxy + exz + fyz
Generally, if x is a column of n variables and A is an n×n matrix then xTAx will be in quadratic form.