converting to an ellipse equation

Question

I have the following question: \begin{equation} z\:=\:x^2+y^2+xy-x+y+1 \end{equation}

after I complete the square, I end up with this \begin{equation} z\:=\left(x-\frac{1}{2}\right)^2+\left(y+\frac{1}{2}\right)^2+xy+\frac{1}{2} \end{equation} however this doesn't look like the generic equation of an ellipse below by any means: \begin{equation} z\:=\left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2 \end{equation}

can anybody please help! much appreciated in advance.

Answer 1

It's $$z=\left(x+\frac{y}{2}-\frac{1}{2}\right)^2+\frac{3}{4}(y+1)^2$$