How to solve the quadratic form I am a physicist and I have a problem solving this
\begin{equation}
Q(x)=\frac{1}{2}(x,Ax)+(b,x)+c
\end{equation}
In a book it says that:
"The minimum of Q lies at $\bar{x}=-A^{-1}b$ and
\begin{equation}
Q(x)=Q(\bar{x})+\frac{1}{2}((x-\bar{x}),A(x-\bar{x}))
\end{equation}
How do I go to this? What how much is $Q(\bar{x})$?

You have 
$$
\frac{1}{2}((x-\bar{x}),A(x-\bar{x}))=\frac{1}{2}(x,Ax)-(x,A\bar{x})+\frac{1}{2}(\bar{x},A\bar{x})
$$
so that
$$
\tag{1}
\frac{1}{2}((x-\bar{x}),A(x-\bar{x}))+Q(\bar x)=\frac{1}{2}(x,Ax)-(x,A\bar{x})+(\bar{x},A\bar{x})+(b,\bar x)+c.
$$
On the other hand:
$$
(\bar x,A\bar x)+(b,\bar x)=(A^{-1}b,b)-(b,A^{-1}b)=0
\quad\hbox{and}\quad
(x,A\bar{x})=-(x,b),
$$
and plugging these into (1) yields the desired result.
