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Beliebt Algebra >

x^5+4+4i=0

Lösung

x^{5} + 4 + 4 i = 0

Lösung

x = \frac{4 + 10 - 2 5}{2} - i \frac{4 - 10 - 2 5}{2}, x = 1 + i, x = - \frac{4 - 10 - 2 5}{2} + i \frac{4 + 10 - 2 5}{2}, x = - \frac{4 + 10 + 2 5}{2} - i \frac{4 - 10 + 2 5}{2}, x = - \frac{4 - 10 + 2 5}{2} - i \frac{4 + 10 + 2 5}{2}

Schritte zur Lösung

x^{5} + 4 + 4 i = 0

Verschiebe

4 i

auf die rechte Seite

x^{5} + 4 = - 4 i

Verschiebe

4

auf die rechte Seite

x^{5} = - 4 i - 4

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 = 5, a = - 4 i - 4

∣ a ∣ = 42

ar g (a) = \frac{π}{4} - π

x = 542 (cos (\frac{\frac{π}{4} - π + 2 \cdot 0 π}{5}) + i sin (\frac{\frac{π}{4} - π + 2 \cdot 0 π}{5})), x = 542 (cos (\frac{\frac{π}{4} - π + 2 \cdot 1 π}{5}) + i sin (\frac{\frac{π}{4} - π + 2 \cdot 1 π}{5})), x = 542 (cos (\frac{\frac{π}{4} - π + 2 \cdot 2 π}{5}) + i sin (\frac{\frac{π}{4} - π + 2 \cdot 2 π}{5})), x = 542 (cos (\frac{\frac{π}{4} - π + 2 \cdot 3 π}{5}) + i sin (\frac{\frac{π}{4} - π + 2 \cdot 3 π}{5})), x = 542 (cos (\frac{\frac{π}{4} - π + 2 \cdot 4 π}{5}) + i sin (\frac{\frac{π}{4} - π + 2 \cdot 4 π}{5}))

Vereinfache

542 (cos (\frac{\frac{π}{4} - π + 2 \cdot 0 π}{5}) + i sin (\frac{\frac{π}{4} - π + 2 \cdot 0 π}{5})) : \frac{4 + 10 - 2 5}{2} - i \frac{4 - 10 - 2 5}{2}

Vereinfache

542 (cos (\frac{\frac{π}{4} - π + 2 \cdot 1 π}{5}) + i sin (\frac{\frac{π}{4} - π + 2 \cdot 1 π}{5})) : 1 + i

Vereinfache

542 (cos (\frac{\frac{π}{4} - π + 2 \cdot 2 π}{5}) + i sin (\frac{\frac{π}{4} - π + 2 \cdot 2 π}{5})) : - \frac{4 - 10 - 2 5}{2} + i \frac{4 + 10 - 2 5}{2}

Vereinfache

542 (cos (\frac{\frac{π}{4} - π + 2 \cdot 3 π}{5}) + i sin (\frac{\frac{π}{4} - π + 2 \cdot 3 π}{5})) : - \frac{4 + 10 + 2 5}{2} - i \frac{4 - 10 + 2 5}{2}

Vereinfache

542 (cos (\frac{\frac{π}{4} - π + 2 \cdot 4 π}{5}) + i sin (\frac{\frac{π}{4} - π + 2 \cdot 4 π}{5})) : - \frac{4 - 10 + 2 5}{2} - i \frac{4 + 10 + 2 5}{2}

x = \frac{4 + 10 - 2 5}{2} - i \frac{4 - 10 - 2 5}{2}, x = 1 + i, x = - \frac{4 - 10 - 2 5}{2} + i \frac{4 + 10 - 2 5}{2}, x = - \frac{4 + 10 + 2 5}{2} - i \frac{4 - 10 + 2 5}{2}, x = - \frac{4 - 10 + 2 5}{2} - i \frac{4 + 10 + 2 5}{2}

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