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

解

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

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}

解答ステップ

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

i x

を左側に移動します

x^{4} = - i + 3

z^{n} = a

の場合, 解は

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

以下のため:

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}))

簡素化

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}

簡素化

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}

簡素化

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}

簡素化

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}