|3z+2i|=3

Lösung

∣ 3 z + 2 i ∣ = 3

z = x + \frac{- 2 + 3 - x ^{2} + 1}{3} i, z = x - \frac{3 - x ^{2} + 1 + 2}{3} i; x \in R

Schritte zur Lösung

∣ 3 z + 2 i ∣ = 3

Ersetze

z = x + y i

∣ 3 (x + y i) + 2 i ∣ = 3

Schreibe

∣ 3 (x + y i) + 2 i ∣

um:

9 x^{2} + 9 y^{2} + 12 y + 4

9 x^{2} + 9 y^{2} + 12 y + 4 = 3

Löse nach

y : y = \frac{- 2 + 3 - x ^{2} + 1}{3}, y = - \frac{3 - x ^{2} + 1 + 2}{3}

Setze in

z = x + y i

ein

z = x + \frac{- 2 + 3 - x ^{2} + 1}{3} i, z = x - \frac{3 - x ^{2} + 1 + 2}{3} i; x \in R

(x + 4 i)^{2} = 3 - 4 i

x^{2} + 2 i x y - y^{2} = 0 + 2 i

x - i y = \frac{7 - i ^{9}}{1 - i}

i (x + i y) = 2

i (x) = \frac{x ^{2}}{4} + 3450 x