Type 1: $z^n$ = w (find all roots) Use polar form: z = |w|^$\frac{1}{n}$ * e^(i*(arg(w) + 2k*pi)/n), k = 0, 1, ..., n-1

Type 2: |z - z1| = |z - z2| (locus) Square both sides: (x-a1)^2 + (y-b1)^2 = (x-a2)^2 + (y-b2)^2, simplifies to a linear equation (line).

Type 3: $z^2$ + az + b = 0 (quadratic in z) Use quadratic formula: z = (-a +/- sqrt($a^2$ - 4b))/2. If discriminant is negative, express sqrt of negative as i*sqrt(positive).

Type 4: Simultaneous equations with z and z-bar Write z = x + iy and z-bar = x - iy, then separate real and imaginary parts to get two real equations.

Type 5: |f(z)| = g(z) type Often: square both sides to use |z|^2 = z*z-bar, then solve the resulting equation.