I am performing an eigenvalue analysis by hand.
I obtain that $$ \lambda^2_{1,2}(a)=p(a)\pm \sqrt{g(a)},$$
in which $p(a) \in \mathbb{R}$ and $g(a) \in \mathbb{R}$ are second order polynomials depending on a parameter $a\in \mathbb{R}$. Is there a way to quickly determine the interval $I_{+}$ / $I_{-}$ for $a$ in which $\lambda$ has a positive / negative real part?