x^2+2kx+9=0

asked by guest
on Sep 20, 2024 at 9:45 am



You asked:

Investigate the equation: \({x}^{2} + 2 k x + 9 = 0\).

MathBot Answer:

The graph of the equation \({x}^{2} + 2 k x + 9 = 0\) is a hyperbola.

The coordinates of its foci are: \(\left(- \frac{\sqrt{5} \sqrt{- \frac{1}{- \frac{\sqrt{5}}{18} - \frac{1}{18}} + \frac{1}{- \frac{1}{18} + \frac{\sqrt{5}}{18}}}}{2 \sqrt{\frac{\sqrt{5}}{2} + \frac{5}{2}}} - \frac{\sqrt{- \frac{1}{- \frac{\sqrt{5}}{18} - \frac{1}{18}} + \frac{1}{- \frac{1}{18} + \frac{\sqrt{5}}{18}}}}{2 \sqrt{\frac{\sqrt{5}}{2} + \frac{5}{2}}}, \frac{\sqrt{- \frac{1}{- \frac{\sqrt{5}}{18} - \frac{1}{18}} + \frac{1}{- \frac{1}{18} + \frac{\sqrt{5}}{18}}}}{\sqrt{\frac{\sqrt{5}}{2} + \frac{5}{2}}}\right)\) and \(\left(\frac{\sqrt{- \frac{1}{- \frac{\sqrt{5}}{18} - \frac{1}{18}} + \frac{1}{- \frac{1}{18} + \frac{\sqrt{5}}{18}}}}{2 \sqrt{\frac{\sqrt{5}}{2} + \frac{5}{2}}} + \frac{\sqrt{5} \sqrt{- \frac{1}{- \frac{\sqrt{5}}{18} - \frac{1}{18}} + \frac{1}{- \frac{1}{18} + \frac{\sqrt{5}}{18}}}}{2 \sqrt{\frac{\sqrt{5}}{2} + \frac{5}{2}}}, - \frac{\sqrt{- \frac{1}{- \frac{\sqrt{5}}{18} - \frac{1}{18}} + \frac{1}{- \frac{1}{18} + \frac{\sqrt{5}}{18}}}}{\sqrt{\frac{\sqrt{5}}{2} + \frac{5}{2}}}\right)\).

The coordinates of its vertices are: \(\left(- \frac{\sqrt{5}}{2 \sqrt{- \frac{1}{18} + \frac{\sqrt{5}}{18}} \sqrt{\frac{\sqrt{5}}{2} + \frac{5}{2}}} - \frac{1}{2 \sqrt{- \frac{1}{18} + \frac{\sqrt{5}}{18}} \sqrt{\frac{\sqrt{5}}{2} + \frac{5}{2}}}, \frac{1}{\sqrt{- \frac{1}{18} + \frac{\sqrt{5}}{18}} \sqrt{\frac{\sqrt{5}}{2} + \frac{5}{2}}}\right)\) and \(\left(\frac{1}{2 \sqrt{- \frac{1}{18} + \frac{\sqrt{5}}{18}} \sqrt{\frac{\sqrt{5}}{2} + \frac{5}{2}}} + \frac{\sqrt{5}}{2 \sqrt{- \frac{1}{18} + \frac{\sqrt{5}}{18}} \sqrt{\frac{\sqrt{5}}{2} + \frac{5}{2}}}, - \frac{1}{\sqrt{- \frac{1}{18} + \frac{\sqrt{5}}{18}} \sqrt{\frac{\sqrt{5}}{2} + \frac{5}{2}}}\right)\).

The asymptotes have equations: \(k \left(- \frac{18}{\sqrt{- \frac{1}{18} + \frac{\sqrt{5}}{18}} \sqrt{\frac{\sqrt{5}}{2} + \frac{5}{2}}} - \frac{9 \sqrt{5}}{\sqrt{\frac{1}{18} + \frac{\sqrt{5}}{18}} \sqrt{\frac{\sqrt{5}}{2} + \frac{5}{2}}} - \frac{9}{\sqrt{\frac{1}{18} + \frac{\sqrt{5}}{18}} \sqrt{\frac{\sqrt{5}}{2} + \frac{5}{2}}}\right) + x \left(- \frac{9 \sqrt{5}}{\sqrt{- \frac{1}{18} + \frac{\sqrt{5}}{18}} \sqrt{\frac{\sqrt{5}}{2} + \frac{5}{2}}} - \frac{9}{\sqrt{- \frac{1}{18} + \frac{\sqrt{5}}{18}} \sqrt{\frac{\sqrt{5}}{2} + \frac{5}{2}}} + \frac{18}{\sqrt{\frac{1}{18} + \frac{\sqrt{5}}{18}} \sqrt{\frac{\sqrt{5}}{2} + \frac{5}{2}}}\right) = 0\) and \(k \left(- \frac{9 \sqrt{5}}{\sqrt{\frac{1}{18} + \frac{\sqrt{5}}{18}} \sqrt{\frac{\sqrt{5}}{2} + \frac{5}{2}}} - \frac{9}{\sqrt{\frac{1}{18} + \frac{\sqrt{5}}{18}} \sqrt{\frac{\sqrt{5}}{2} + \frac{5}{2}}} + \frac{18}{\sqrt{- \frac{1}{18} + \frac{\sqrt{5}}{18}} \sqrt{\frac{\sqrt{5}}{2} + \frac{5}{2}}}\right) + x \left(\frac{9}{\sqrt{- \frac{1}{18} + \frac{\sqrt{5}}{18}} \sqrt{\frac{\sqrt{5}}{2} + \frac{5}{2}}} + \frac{18}{\sqrt{\frac{1}{18} + \frac{\sqrt{5}}{18}} \sqrt{\frac{\sqrt{5}}{2} + \frac{5}{2}}} + \frac{9 \sqrt{5}}{\sqrt{- \frac{1}{18} + \frac{\sqrt{5}}{18}} \sqrt{\frac{\sqrt{5}}{2} + \frac{5}{2}}}\right) = 0\).

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