Originally Posted by
skeeter Maybe I'm missing something in the question (see attached graph of $y=\tan^2{x}-\tan{x}$) ...
For $k < -\dfrac{1}{4}$, $\tan^2{x}-\tan{x} = k$ has no real solutions.
For $k \ge -\dfrac{1}{4}$, $\tan^2{x}-\tan{x}=k$ has an infinite number of real solutions.
If we're considering only a single period of the function ...
$k < -\dfrac{1}{4}$ yields no solutions, $k = -\dfrac{1}{4}$ yields a single solution, and $k > - \dfrac{1}{4}$ yields only two solutions.
Sorry, but I'm not seeing how the equation yields three real solutions unless there is a restricted interval for $x$ that was unstated in the original problem ... someone please enlighten me.