I was solving a normal-mode problem and got a different result for the quadratic equation. The book provides a simpler solution than mine so I suspect I am the one who's wrong. Let's check it out.
Let us start from the following determinant
$$ \begin{vmatrix} \omega_o^2-\omega^2 & \frac{-ieB\omega}{m} \\ \frac{ieB\omega}{m} & \omega_o^2-\omega^2 \\ \end{vmatrix}= (\omega_o^2-\omega^2)^2-\Big(\frac{eB\omega}{m}\Big)^2=\omega^4-\Big[\Big(\frac{eB}{m}\Big)^2+2\omega_0^2\Big]\omega^2+\omega_0^4 $$
OK so far.
From here on I proceeded as follows; I looked for the roots, i.e. $\omega^4-\Big[\Big(\frac{eB}{m}\Big)^2+2\omega_0^2\Big]\omega^2+\omega_0^4=0$
$$\omega^2= \frac 1 2 \Big[\Big(\frac{eB}{m}\Big)^2+2\omega_0^2 \pm\ \sqrt{\Big[\Big(\frac{eB}{m}\Big)^2+2\omega_0^2\Big]^2-4\omega_0^4}\Big]$$
This leads to pretty ugly roots for $\omega$.
However, the book states that $(\omega_o^2-\omega^2)^2-\Big(\frac{eB\omega}{m}\Big)^2$ leads to $\omega^2 \pm \frac{eB\omega}{m} - \omega_o^2$. This leads to good looking roots for $\omega$.
My struggle is that I do not see how to show that's indeed the case.