Identify the conic section $$$1 - 2 x^{2} = x - 1$$$

Identify and find the properties of the conic section $$$1 - 2 x^{2} = x - 1$$$.

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, $$$A = 2$$$, $$$B = 0$$$, $$$C = 0$$$, $$$D = 1$$$, $$$E = 0$$$, $$$F = -2$$$.

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.

$$$1 - 2 x^{2} = x - 1$$$A represents a pair of the lines $$$x = - \frac{1 + \sqrt{17}}{4}$$$, $$$x = \frac{-1 + \sqrt{17}}{4}$$$A.

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

Factored form: $$$\left(4 x + 1 + \sqrt{17}\right) \left(4 x - \sqrt{17} + 1\right) = 0$$$A.

Graph: see the graphing calculator.