Identify the conic section $$$\frac{x^{2}}{6} - \frac{x}{4} = 6$$$

Identify and find the properties of the conic section $$$\frac{x^{2}}{6} - \frac{x}{4} = 6$$$.

The general equation of a conic section is $$$A x^{2} + B x y + C y^{2} + D x + E y + F = 0$$$.

In our case, $$$A = \frac{1}{6}$$$, $$$B = 0$$$, $$$C = 0$$$, $$$D = - \frac{1}{4}$$$, $$$E = 0$$$, $$$F = -6$$$.

The discriminant of the conic section is $$$\Delta = 4 A C F - A E^{2} - B^{2} F + B D E - C D^{2} = 0$$$.

Next, $$$B^{2} - 4 A C = 0$$$.

Since $$$\Delta = 0$$$, this is the degenerated conic section.

Since $$$B^{2} - 4 A C = 0$$$, the equation represents two parallel lines.

$$$\frac{x^{2}}{6} - \frac{x}{4} = 6$$$A represents a pair of the lines $$$x = - \frac{3 \left(-1 + \sqrt{65}\right)}{4}$$$, $$$x = \frac{3 \left(1 + \sqrt{65}\right)}{4}$$$A.

General form: $$$\frac{x^{2}}{6} - \frac{x}{4} - 6 = 0$$$A.

Factored form: $$$\left(4 x - 3 + 3 \sqrt{65}\right) \left(4 x - 3 \sqrt{65} - 3\right) = 0$$$A.

Graph: see the graphing calculator.