Conics

Complete square to get from non-standard form to standard to graph

Example

$\begin{aligned} x^{2} + 2 y^{2} + 6 x - 4 y + 7 & = 0 \\ x^{2} + 6 x + 9 - 9 + 2 (y^{2} - 2 y + 1 - 1) + 7 & = 0 \\ (x + 3)^{2} + 2 (y - 1)^{2} - 4 & = 0 \\ \frac{(x + 3)^{2}}{4} + \frac{(y - 1)^{2}}{2} = 1 \end{aligned}$

centre: (-3, 1)