How do I solve quadratic equations when the coefficients are complex and real? I needed to solve this: $$x^2 + (2i-3)x + 2-4i = 0 $$
I tried the quadratic formula but it didn't work. So how do I solve this without "guessing" roots? If I guess $x=2$ it works; then I can divide the polynomial and find the other root; but I can't "guess" a root.
$b^2-4ac=4i-3$, now I have to work with $\sqrt{4-3i}$ which I don't know how. Apparently $4i-3$ is equal to $(1+2i)^2$, but I don't know how to get to this answer, so I am stuck.

You can find real roots (or purely imaginary roots) without guessing, and in some ways it's nicer than finding square roots of complex discriminants. A real root will preserve the real and complex components, so just write down the equations along the real and imaginary axes. In this case
$$
x^2-3x+2=0 \quad \quad 2ix -4i = 0
$$
$2$ satisfies both so is a root.
