So I have this quadratic of the form:
$z^2 - \frac{e^{i\theta} + 1}{\overline{a}}z + \frac{a}{\overline{a}}e^{i\theta},$
where a is in the open unit disk. I'm trying to find values for a and theta which will allow me to factor it into the form $(z-e^{i\phi})(z-e^{i\psi})$ where $e^{i\phi} \neq e^{i\psi}$. My basic approach has been to set:
$e^{i\phi} + e^{i\psi} = \frac{e^{i\theta} + 1}{\overline{a}},$
$e^{i\phi}e^{i\psi} = \frac{a}{\overline{a}}e^{i\theta},$
and then try to solve for the various variables, but there are just too many unknowns, I can't find a systematic way to do this, hopefully someone can help me with this, thanks.