Geometric difference between $x^TAx$ and $x^TAx + b^Tx + c$ What is the difference between $x^TAx$ and $x^TAx + b^Tx + c$ geometrically? Some analogous examples from quadriatic equations would be great.

Simply, the first case is an homogeneous polynomial of degree 2; the second is a non homogeneous polynomial of degree 2.
EDIT: Maybe I can't really understand your question. I hope this comment can be helpful. Consider that if $x \in \mathbb{R}^{n}$ you can rewrite ${x}^{T} A x + {b}^{T} x +c$ in this way: $(x^{T},1) B (x^{T},1)^{T}$ where B is $\left(\begin{array}{c|c} A & b^{T} \\ \hline 0 & c \end{array}\right)$
