Identify the conic section $$$21 x^{2} - 32 x - 5 = 0$$$

Identify and find the properties of the conic section $$$21 x^{2} - 32 x - 5 = 0$$$.

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, $$$A = 21$$$, $$$B = 0$$$, $$$C = 0$$$, $$$D = -32$$$, $$$E = 0$$$, $$$F = -5$$$.

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.

$$$21 x^{2} - 32 x - 5 = 0$$$A represents a pair of the lines $$$x = - \frac{1}{7}$$$, $$$x = \frac{5}{3}$$$A.

General form: $$$21 x^{2} - 32 x - 5 = 0$$$A.

Factored form: $$$\left(3 x - 5\right) \left(7 x + 1\right) = 0$$$A.

Graph: see the graphing calculator.