Can someone help me find the exact solutions to the equation
I've used a grapher to find approximate solutions at the three places where it crosses the x-axis, and tried my best to work through Cardano's solution ( ) but I just can't finish it.
A big part of my problem here is that using the rational root theorem leads me to try , none of which work, so all three real solutions (it crosses three times, so no complex-imaginary roots) must be irrational. I recognize it obviously must be possible, but I'm not used to three irrational roots giving such a nice, simplistic equation. These three irrationals must multiply to be perfectly 3, add to be perfectly -2, and something else (what?) to be perfectly -9 (or +9? I can't remember).
Obviously I've put a good amount of thought into this problem; somebody please help me out!