I'm trying to factor:
$ 4xy-8y^2-2x^2+9x=0$
The equation is an ellipse so it should be possible to get this into the form $\frac{(x-a)^2}{p} + \frac{(y-b)^2}{q} - c=0$
I've tried "completing the square" in several ways but I can't find any nice way to deal with the $4xy$ term since it has both variables. Thanks!
The ellipse writes in matrix form $$ (\mathbf{x-c})^T \mathbf{A} (\mathbf{x-c}) = \mathbf{x}^T \mathbf{A} \mathbf{x} - 2\mathbf{c}^T \mathbf{A} \mathbf{x} + \mathbf{c}^T \mathbf{A} \mathbf{c} = R^2 $$ From the equation you give, by identification $$ \mathbf{A}= \begin{pmatrix} 2 & -2 \\ -2 & 8 \end{pmatrix}, \mathbf{c} = \begin{pmatrix} 3 \\ 3/4 \end{pmatrix}, $$ and thus $R^2=27/2$.