I forgot to put the link, sorry
http://www.stewartcalculus.com/data/...Orders_Stu.pdf
They probably wrote the imaginary part as
$\displaystyle \frac{\sqrt{-c^2 + 4mk}}{2m}$
This is actually correct since the imaginary pary is not going to have the $\displaystyle i = \sqrt{-1}$. Thus we have to factor this out of the determinant.
If $\displaystyle c^2>4mk$ the solution would not be imaginary. Notice the way I have it written already assumes that we have an imaginary solution, and I have factored out the $\displaystyle i$. That is why it is $\displaystyle -c^2 + 4mk$ instead of $\displaystyle c^2 - 4mk$.