x^3+8i=0

Lösung

x^{3} + 8 i = 0

x = 3 - i, x = 2 i, x = - 3 - i

Schritte zur Lösung

x^{3} + 8 i = 0

Verschiebe

8 i

auf die rechte Seite

x^{3} = - 8 i

Für

z^{n} = a

sind die Lösungen

z_{k} = n ∣ a ∣ (cos (\frac{ar g ( a ) + 2 kπ}{n}) + i sin (\frac{ar g ( a ) + 2 kπ}{n})),

k = 0, 1, \dots, n - 1

Für

n = 3, a = - 8 i

∣ a ∣ = 8

ar g (a) = - \frac{π}{2}

x = 38 (cos (\frac{- \frac{π}{2} + 2 \cdot 0 π}{3}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 0 π}{3})), x = 38 (cos (\frac{- \frac{π}{2} + 2 \cdot 1 π}{3}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 1 π}{3})), x = 38 (cos (\frac{- \frac{π}{2} + 2 \cdot 2 π}{3}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 2 π}{3}))

Vereinfache

38 (cos (\frac{- \frac{π}{2} + 2 \cdot 0 π}{3}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 0 π}{3})) : 3 - i

Vereinfache

38 (cos (\frac{- \frac{π}{2} + 2 \cdot 1 π}{3}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 1 π}{3})) : 2 i

Vereinfache

38 (cos (\frac{- \frac{π}{2} + 2 \cdot 2 π}{3}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 2 π}{3})) : - 3 - i

x = 3 - i, x = 2 i, x = - 3 - i

so l v e f or x, x^{3} - x^{2} + x - 2 = 4

(x^{2} + x)^{2} - 18 (x^{2} + x) + 72 = 0

(w - 3) (w^{2} + 3 w + 9) = 2

x^{6} - 5 x^{4} + 6 x^{2} - 1 = 0

- x^{3} - 11 x^{2} - 39 x - 45 = 0