Originally Posted by

**icemanfan** The equation is a hyperbola because the $\displaystyle x^2$ and $\displaystyle y^2$ terms on the same side of the equation have opposite sign. Also, you made some mistakes. Let me show you where:

$\displaystyle x^2 - 4x - 4y^2 - 24y = 48$ is perfect.

$\displaystyle x^2 - 4x + 4 - 4(y^2 + 6y + 9) = 48 + 4 - 36$ is the next step.

$\displaystyle (x - 2)^2 - 4(y + 3)^2 = 16$

$\displaystyle \frac{(x-2)^2}{16} - \frac{(y+3)^2}{4} = 1$

Also, you FOILED $\displaystyle (x-2)(x+2)$ incorrectly. The $\displaystyle 2x$ terms "cancel out" and you have a difference of squares: $\displaystyle x^2 - 4$.