Solve the following for z : $z^{2} - (3 - 2 i) z = (5 i - 5)$

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Solution

$z^{2} - (3 - 2 i) z - (5 i - 5) = 0 z = \frac{(3 - 2 i) \pm \sqrt{9 - 4 - 12 i + 4 (5 i - 5)}}{2} z = \frac{(3 - 2 i) \pm \sqrt{5 - 12 i + 20 i - 20}}{2} z = \frac{(3 - 2 i) \pm \sqrt{- 15 + 8 i}}{2} z = \frac{(3 - 2 i) \pm i \sqrt{15 - 8 i}}{2} l e t, \sqrt{15 - 8 i} = x + i y 15 - 8 i = x^{2} - x^{2} - y^{2} + 2 x y i x^{2} - y^{2} = 15 2 x y = - 8 x y = - 4 y = - \frac{4}{x} x^{2} - \frac{16}{x^{2}} = 15 x^{4} - {15 x}^{2} - 16 = 0 x^{4} - 16 x^{2} + x^{2} - 16 = 0 (x^{2} + 1) (x^{2} - 16) = 0 x = 4; y = - 1 \sqrt{15 - 8 i} = 4 - i \frac{(3 - 2 i) \pm i (4 - i)}{2} \frac{(3 - 2 i) \pm (1 - 4 i)}{2}$

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