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Beliebt Algebra >

x^4+ix=ix-i+sqrt(3)

Lösung

x^{4} + i x = i x - i + 3

Lösung

x = \frac{2 ^{\frac{3}{4}} 4 + 2 + 6}{4} - i \frac{2 ^{\frac{3}{4}} 4 - 2 - 6}{4}, x = \frac{2 ^{\frac{3}{4}} 4 - 2 - 6}{4} + i \frac{2 ^{\frac{3}{4}} 4 + 2 + 6}{4}, x = - \frac{2 ^{\frac{3}{4}} 4 + 2 + 6}{4} + i \frac{2 ^{\frac{3}{4}} 4 - 2 - 6}{4}, x = - \frac{2 ^{\frac{3}{4}} 4 - 2 - 6}{4} - i \frac{2 ^{\frac{3}{4}} 4 + 2 + 6}{4}

Schritte zur Lösung

x^{4} + i x = i x - i + 3

Verschiebe

i x

auf die linke Seite

x^{4} = - i + 3

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 = 4, a = - i + 3

∣ a ∣ = 2

ar g (a) = - \frac{π}{6}

x = 42 (cos (\frac{- \frac{π}{6} + 2 \cdot 0 π}{4}) + i sin (\frac{- \frac{π}{6} + 2 \cdot 0 π}{4})), x = 42 (cos (\frac{- \frac{π}{6} + 2 \cdot 1 π}{4}) + i sin (\frac{- \frac{π}{6} + 2 \cdot 1 π}{4})), x = 42 (cos (\frac{- \frac{π}{6} + 2 \cdot 2 π}{4}) + i sin (\frac{- \frac{π}{6} + 2 \cdot 2 π}{4})), x = 42 (cos (\frac{- \frac{π}{6} + 2 \cdot 3 π}{4}) + i sin (\frac{- \frac{π}{6} + 2 \cdot 3 π}{4}))

Vereinfache

42 (cos (\frac{- \frac{π}{6} + 2 \cdot 0 π}{4}) + i sin (\frac{- \frac{π}{6} + 2 \cdot 0 π}{4})) : \frac{2 ^{\frac{3}{4}} 4 + 2 + 6}{4} - i \frac{2 ^{\frac{3}{4}} 4 - 2 - 6}{4}

Vereinfache

42 (cos (\frac{- \frac{π}{6} + 2 \cdot 1 π}{4}) + i sin (\frac{- \frac{π}{6} + 2 \cdot 1 π}{4})) : \frac{2 ^{\frac{3}{4}} 4 - 2 - 6}{4} + i \frac{2 ^{\frac{3}{4}} 4 + 2 + 6}{4}

Vereinfache

42 (cos (\frac{- \frac{π}{6} + 2 \cdot 2 π}{4}) + i sin (\frac{- \frac{π}{6} + 2 \cdot 2 π}{4})) : - \frac{2 ^{\frac{3}{4}} 4 + 2 + 6}{4} + i \frac{2 ^{\frac{3}{4}} 4 - 2 - 6}{4}

Vereinfache

42 (cos (\frac{- \frac{π}{6} + 2 \cdot 3 π}{4}) + i sin (\frac{- \frac{π}{6} + 2 \cdot 3 π}{4})) : - \frac{2 ^{\frac{3}{4}} 4 - 2 - 6}{4} - i \frac{2 ^{\frac{3}{4}} 4 + 2 + 6}{4}

x = \frac{2 ^{\frac{3}{4}} 4 + 2 + 6}{4} - i \frac{2 ^{\frac{3}{4}} 4 - 2 - 6}{4}, x = \frac{2 ^{\frac{3}{4}} 4 - 2 - 6}{4} + i \frac{2 ^{\frac{3}{4}} 4 + 2 + 6}{4}, x = - \frac{2 ^{\frac{3}{4}} 4 + 2 + 6}{4} + i \frac{2 ^{\frac{3}{4}} 4 - 2 - 6}{4}, x = - \frac{2 ^{\frac{3}{4}} 4 - 2 - 6}{4} - i \frac{2 ^{\frac{3}{4}} 4 + 2 + 6}{4}

Beliebte Beispiele

x^4+6*x^2-60x+36=0

x^{4} + 6 \cdot x^{2} - 60 x + 36 = 0

x^{4} - 6 x^{2} = 27

x (x - 3) (2 x - 1) = 0

j(x)=3x^3,[1,1+h(x)]

j (x) = 3 x^{3}, [1, 1 + h (x)]

6 x^{3} - 2 = 0