Is there a quick way to find this matrix A? I want to find a matrix $\mathbf A$ such that $x^2 + 4y^2 + 9z^2 + 4xy - 6xz = 1$ can be written as 
$$\begin{bmatrix}x&y&z\end{bmatrix}\mathbf A\begin{bmatrix}x&y&z\end{bmatrix}^\top = 1$$
Is there a quick way to do this? I can tell $\mathbf A$ is
$$\begin{bmatrix}
    1 & x_{12} & x_{13}\\
    x_{21} &4 & x_{23}\\
    x_{31} & x_{32} & 9
\end{bmatrix}$$
But I need to guess the other elements. Is there a quick way to solve this?

Let consider


*

*$x_{12}=x_{21}=2$

*$x_{13}=x_{31}=-3$

*$x_{23}=x_{32}=0$


and in general
$$ax^2 + by^2 + cz^2 + 2dxy+2exz+2fyz\implies  A=\begin{bmatrix}
    a & d & e\\
    d & b & f\\
    e & f & c
\end{bmatrix}$$
