a quadratic form's question

a quadratic form's question

a quadratic form



a curve
$$\begin{align*}\text{ax}^2+\text{bxy}+\text{cy}^2+\text{dx}+\text{ey}+f\tag{1}\end{align*}$$
$$\begin{align*}\text{ax}^2+2\text{bxy}+\text{cy}^2+2\text{dx}+2\text{ey}+f\tag{2}\end{align*}$$
the matrix of (1) is
$$\begin{align*}\begin{array}{ccc} a & \frac{b}{2} & \frac{d}{2} \\
\frac{b}{2} & c & \frac{e}{2} \\ \frac{d}{2} & \frac{e}{2} & f
\\\end{array}\end{align*}$$
the matrix of (2) is
$$\begin{align*}\begin{array}{ccc} a & b & d \\ b & c & e \\ d & e & f
\\\end{array}\end{align*}$$
$b=\frac{b}{2}+\frac{b}{2}$, question 1: what's the meaning of the
addition in Matrix?
I know the Trace is the summation of elements of the dominant diagonal.
question 2: What's the realtionship between (1) and (2)