How do I find the roots of this polynomial with complex coefficients:
p(z) = z^{2}-(1+2i)z + (- (3/2)+2i)
Is there a general method?
yes - the quadratic equation.
Square roots of complex numbers are generally calculated by putting them into polar form first:
$\displaystyle x + iy = \sqrt{x^2 + y^2}e^{i\tan^{-1}(y/x)}$ when $\displaystyle x>0$,
$\displaystyle x + iy = \sqrt{x^2 + y^2}e^{i(\pi + \tan^{-1}(y/x))}$ when $\displaystyle x<0$,
$\displaystyle x + iy = \sqrt{x^2 + y^2}e^{i \pi/2 }$ when $\displaystyle x=0, y>0$,
$\displaystyle x + iy = \sqrt{x^2 + y^2}e^{- i \pi/2 }$ when $\displaystyle x=0, y<0$.