Identify the conic section $$$x + 45 = x \left(\frac{11 x}{20} + \frac{231}{5}\right)$$$

Identify and find the properties of the conic section $$$x + 45 = x \left(\frac{11 x}{20} + \frac{231}{5}\right)$$$.

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, $$$A = \frac{11}{20}$$$, $$$B = 0$$$, $$$C = 0$$$, $$$D = \frac{226}{5}$$$, $$$E = 0$$$, $$$F = -45$$$.

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.

$$$x + 45 = x \left(\frac{11 x}{20} + \frac{231}{5}\right)$$$A represents a pair of the lines $$$x = - \frac{2 \left(226 + \sqrt{53551}\right)}{11}$$$, $$$x = \frac{2 \left(-226 + \sqrt{53551}\right)}{11}$$$A.

General form: $$$\frac{11 x^{2}}{20} + \frac{226 x}{5} - 45 = 0$$$A.

Factored form: $$$\left(11 x + 452 + 2 \sqrt{53551}\right) \left(11 x - 2 \sqrt{53551} + 452\right) = 0$$$A.

Graph: see the graphing calculator.