Quadratic forms (Reduced forms) Problem: Find a reduced form equivalent to $7x^2 + 25xy + 23y^2$
I know there are two forms to do it, when $c<a$ and $|b|>a$ I did the process but I don't know how to do the matrix. 
$$7x^2 + 25xy + 23y^2\quad (|b|>a)$$
and $D=b^2 -4ac =-19$ doing the formula I have $k=-2$, $b'=9$ and $c'=1$
The result it will be $7x^2 -3xy + y^2$, I have all the sketch with the formulas (Next, $x^2 + 3xy + 7y^2$ and $x^2 + xy + 5y^2$), but none of the matrix 
Thanks for your help
 A: I prefer to write the matrices on the right...
The mapping
$$  \langle a, b, c \rangle  \mapsto  \langle a, \; b + 2at, \; c +bt +at^2 \rangle   $$
is brought about by the matrix
$$ 
\left(
\begin{array}{rr}
1 & t \\
0 & 1
\end{array}
\right)
$$
The mapping
$$  \langle a, b, c \rangle  \mapsto  \langle c, -b , a \rangle   $$
is brought about by the matrix
$$ 
\left(
\begin{array}{rr}
0 & -1 \\
1 & 0
\end{array}
\right)
$$
==================================================
$$  \langle 7, 25, 23 \rangle $$
$$ 
R_1 = 
\left(
\begin{array}{rr}
1 & -2 \\
0 & 1
\end{array}
\right)
$$
$$  \langle 7, -3, 1 \rangle $$
$$  
R_2 =
\left(
\begin{array}{rr}
0 & -1 \\
1 & 0
\end{array}
\right)
$$
$$  \langle 1, 3, 7 \rangle $$
$$ 
R_3 = 
\left(
\begin{array}{rr}
1 & -1 \\
0 & 1
\end{array}
\right)
$$
$$  \langle 1, 1, 5 \rangle $$
$$ 
R = R_1 R_2 R_3 =
\left(
\begin{array}{rr}
-2 & 1 \\
1 & -1
\end{array}
\right)
$$
$$ 
H =  
\left(
\begin{array}{rr}
14 & 25 \\
25 & 46
\end{array}
\right)
$$
$$ 
G =  
\left(
\begin{array}{rr}
2 & 1 \\
1 & 10
\end{array}
\right)
$$
does give
$$ R^T H R = G $$
