Quadratic equation with complex variable

Question

I need to solve the following equation with $\omega \in \mathbb{C}$ and $\gamma, \omega_0 \in \mathbb{R}$ for its complex roots $\omega_1, \omega_2$ $$ \omega^2 + \gamma \omega + \omega_0^2 = 0 $$

I've tried writing $\omega = x+iy$ but it seems to only make the equation more complicated. Because the variable $\omega$ is complex, I assume I can't use the standard quadratic formula. Should I just complete the square or even use $\omega$'s polar representation?

This seems like a very easy problem but complex equations don't click in my head yet and resources I have found on the internet just deal with real-valued equations that have complex solutions.

Thanks!

Answer 1

The quadratic formula applies also to cases where the roots are complex. $$\omega = \frac{-\gamma\pm \sqrt{\gamma^2 -4\omega_0^2}}{2} $$

Depending on the sign of the $\gamma^2 -4\omega_0^2$, $\omega$ may be imaginary/real.

Answer 2

Well: $$(x+iy)^2+\gamma(x+iy)+z^2=0\implies x^2-y^2+2xyi+\gamma x + i\gamma y +z^2=0$$

$$\implies x^2-y^2+x+z^2=0; iy(2x+\gamma)=0$$ by separating the real and imaginary parts.

Two cases are seen here: either $y=0$ (so $\omega$ is not complex) or $x=-\frac \gamma 2$, from which we solve the quadratic in $y$.