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Beliebt Algebra >

z^6+2z^3+2=0

Lösung

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

Lösung

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

Schritte zur Lösung

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

Schreibe die Gleichung um mit

u = z^{3}

und

u^{2} = z^{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 = z^{3} w i e d er e in,

löse für

z

Löse

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

Löse

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

Die Lösungen sind

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

Graph

Beliebte Beispiele

a^{4} = 54

- x^{3} + x^{2} + x + 1 = 0

z^{6} - 8 = 0

0 = x^{3} + 4 x - 5

0 = x^{4} - 2 x^{2}