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Identify the conic section $$$343 x \left(3 x - 16\right) = 4802$$$
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Your Input
Identify and find the properties of the conic section $$$343 x \left(3 x - 16\right) = 4802$$$.
Solution
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, $$$A = 1029$$$, $$$B = 0$$$, $$$C = 0$$$, $$$D = -5488$$$, $$$E = 0$$$, $$$F = -4802$$$.
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.
Answer
$$$343 x \left(3 x - 16\right) = 4802$$$A represents a pair of the lines $$$x = - \frac{-8 + \sqrt{106}}{3}$$$, $$$x = \frac{8 + \sqrt{106}}{3}$$$A.
General form: $$$1029 x^{2} - 5488 x - 4802 = 0$$$A.
Factored form: $$$\left(3 x - 8 + \sqrt{106}\right) \left(3 x - \sqrt{106} - 8\right) = 0$$$A.
Graph: see the graphing calculator.