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Identify the conic section $$$64 x^{2} - 144 x + 81 = 0$$$
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Your Input
Identify and find the properties of the conic section $$$64 x^{2} - 144 x + 81 = 0$$$.
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, $$$A = 64$$$, $$$B = 0$$$, $$$C = 0$$$, $$$D = -144$$$, $$$E = 0$$$, $$$F = 81$$$.
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 a line.
Answer
$$$64 x^{2} - 144 x + 81 = 0$$$A represents the line $$$x = \frac{9}{8}$$$A.
General form: $$$64 x^{2} - 144 x + 81 = 0$$$A.
Factored form: $$$\left(8 x - 9\right)^{2} = 0$$$A.
Graph: see the graphing calculator.