Identify the conic section $$$x^{2} - 4 x = 10$$$

Identify and find the properties of the conic section $$$x^{2} - 4 x = 10$$$.

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, $$$A = 1$$$, $$$B = 0$$$, $$$C = 0$$$, $$$D = -4$$$, $$$E = 0$$$, $$$F = -10$$$.

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.

$$$x^{2} - 4 x = 10$$$A represents a pair of the lines $$$x = 2 - \sqrt{14}$$$, $$$x = 2 + \sqrt{14}$$$A.

General form: $$$x^{2} - 4 x - 10 = 0$$$A.

Factored form: $$$\left(x - 2 + \sqrt{14}\right) \left(x - \sqrt{14} - 2\right) = 0$$$A.

Graph: see the graphing calculator.