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Beliebt Algebra >

x^6+2x^3+2=0

Lösung

x^{6} + 2 x^{3} + 2 = 0

Lösung

x = \frac{2 ^{\frac{2}{3}}}{2} + i \frac{2 ^{\frac{2}{3}}}{2}, x = \frac{2 ^{\frac{2}{3}} ( - 1 - 3 )}{4} + i \frac{2 ^{\frac{2}{3}} ( - 1 + 3 )}{4}, x = \frac{2 ^{\frac{2}{3}} ( - 1 + 3 )}{4} - i \frac{2 ^{\frac{2}{3}} ( 1 + 3 )}{4}, x = \frac{2 ^{\frac{2}{3}}}{2} - i \frac{2 ^{\frac{2}{3}}}{2}, x = \frac{2 ^{\frac{2}{3}} ( - 1 + 3 )}{4} + i \frac{2 ^{\frac{2}{3}} ( 1 + 3 )}{4}, x = - \frac{2 ^{\frac{2}{3}} ( 1 + 3 )}{4} - i \frac{2 ^{\frac{2}{3}} ( - 1 + 3 )}{4}

Schritte zur Lösung

x^{6} + 2 x^{3} + 2 = 0

Schreibe die Gleichung um mit

u = x^{3}

und

u^{2} = x^{6}

u^{2} + 2 u + 2 = 0

Löse

u^{2} + 2 u + 2 = 0 : u = - 1 + i, u = - 1 - i

u = - 1 + i, u = - 1 - i

Setze

u = x^{3} w i e d er e in,

löse für

x

Löse

x^{3} = - 1 + i : x = \frac{2 ^{\frac{2}{3}}}{2} + i \frac{2 ^{\frac{2}{3}}}{2}, x = \frac{2 ^{\frac{2}{3}} ( - 1 - 3 )}{4} + i \frac{2 ^{\frac{2}{3}} ( - 1 + 3 )}{4}, x = \frac{2 ^{\frac{2}{3}} ( - 1 + 3 )}{4} - i \frac{2 ^{\frac{2}{3}} ( 1 + 3 )}{4}

Löse

x^{3} = - 1 - i : x = \frac{2 ^{\frac{2}{3}}}{2} - i \frac{2 ^{\frac{2}{3}}}{2}, x = \frac{2 ^{\frac{2}{3}} ( - 1 + 3 )}{4} + i \frac{2 ^{\frac{2}{3}} ( 1 + 3 )}{4}, x = - \frac{2 ^{\frac{2}{3}} ( 1 + 3 )}{4} - i \frac{2 ^{\frac{2}{3}} ( - 1 + 3 )}{4}

Die Lösungen sind

x = \frac{2 ^{\frac{2}{3}}}{2} + i \frac{2 ^{\frac{2}{3}}}{2}, x = \frac{2 ^{\frac{2}{3}} ( - 1 - 3 )}{4} + i \frac{2 ^{\frac{2}{3}} ( - 1 + 3 )}{4}, x = \frac{2 ^{\frac{2}{3}} ( - 1 + 3 )}{4} - i \frac{2 ^{\frac{2}{3}} ( 1 + 3 )}{4}, x = \frac{2 ^{\frac{2}{3}}}{2} - i \frac{2 ^{\frac{2}{3}}}{2}, x = \frac{2 ^{\frac{2}{3}} ( - 1 + 3 )}{4} + i \frac{2 ^{\frac{2}{3}} ( 1 + 3 )}{4}, x = - \frac{2 ^{\frac{2}{3}} ( 1 + 3 )}{4} - i \frac{2 ^{\frac{2}{3}} ( - 1 + 3 )}{4}

Graph

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