Problem:
Given the equation $x^2+6 x-2 y^2-y+8=0$, rewrite and classify the conic and state relevant properties.
Answer:
Classification: Hyperbola
Equation: $(x+3)^2-2 \left(y+\frac{1}{4}\right)^2=\frac{7}{8}$
Foci: $\left(
\begin{array}{cc}
 -3-\frac{\sqrt{21}}{4} & -\frac{1}{4} \\
 \frac{\sqrt{21}}{4}-3 & -\frac{1}{4} \\
\end{array}
\right)$
Eccentricity: $\sqrt{\frac{3}{2}}$
Center: $\left\{-3,-\frac{1}{4}\right\}$
Asymptotes: $\left\{y=\frac{x}{\sqrt{2}}+\frac{1}{4} \left(6 \sqrt{2}-1\right),y=\frac{1}{4} \left(-1-6 \sqrt{2}\right)-\frac{x}{\sqrt{2}}\right\}$
{'source': 'AMPS', 'file': 'amps/mathematica/algebra/conic_sections/6026.txt'}