There are several ways to solve a quadratic equation. I'll complete the square:
z ² - (3 - 2i ) z + 5 - 5i = 0
(z - (3 - 2i )/2)² - ((3 - 2i )/2)² + 5 - 5i = 0
(z - (3 - 2i )/2)² - (3 - 2i )²/4 + 5 - 5i = 0
(z - (3 - 2i )/2)² = (3 - 2i )²/4 - 5 + 5i
(z - (3 - 2i )/2)² = (3² - 2×3×(-2i ) + (-2i )²)/4 - 5 + 5i
(z - (3 - 2i )/2)² = (5 - 12i )/4 - 5 + 5i
(z - (3 - 2i )/2)² = (5 - 12i - 20 + 20i )/4
(z - (3 - 2i )/2)² = (-15 + 8i )/4
Let w = (-15 + 8i )/4. Then write w = |w| exp(i arg(w)), where
|w| = √((-15/4)² + 2²) = 17/4
arg(w) = π - arctan(8/15)
The square roots of w are then
√w = √(|w|) exp(i (arg(w) + 2nπ)/2)
where n in the set {0, 1}.
Taking the square root of both sides gives
z - (3 - 2i )/2 = √w
z = (3 - 2i )/2 + √w
and the two solutions can be simplified to
z = √17/4 + √17 i
z = -√17/4 - √17 i