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Beliebt Algebra >

z^6-2z^3+4=0

Lösung

z^{6} - 2 z^{3} + 4 = 0

Lösung

z = 32 cos (\frac{π}{9}) + 32 i sin (\frac{π}{9}), z = 32 cos (\frac{7 π}{9}) + 32 i sin (\frac{7 π}{9}), z = 32 cos (\frac{13 π}{9}) + 32 i sin (\frac{13 π}{9}), z = 32 cos (\frac{π}{9}) - 32 i sin (\frac{π}{9}), z = 32 cos (\frac{5 π}{9}) + 32 i sin (\frac{5 π}{9}), z = 32 cos (\frac{11 π}{9}) + 32 i sin (\frac{11 π}{9})

Schritte zur Lösung

z^{6} - 2 z^{3} + 4 = 0

Schreibe die Gleichung um mit

u = z^{3}

und

u^{2} = z^{6}

u^{2} - 2 u + 4 = 0

Löse

u^{2} - 2 u + 4 = 0 : u = 1 + 3 i, u = 1 - 3 i

u = 1 + 3 i, u = 1 - 3 i

Setze

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

löse für

z

Löse

z^{3} = 1 + 3 i : z = 32 cos (\frac{π}{9}) + 32 i sin (\frac{π}{9}), z = 32 cos (\frac{7 π}{9}) + 32 i sin (\frac{7 π}{9}), z = 32 cos (\frac{13 π}{9}) + 32 i sin (\frac{13 π}{9})

Löse

z^{3} = 1 - 3 i : z = 32 cos (\frac{π}{9}) - 32 i sin (\frac{π}{9}), z = 32 cos (\frac{5 π}{9}) + 32 i sin (\frac{5 π}{9}), z = 32 cos (\frac{11 π}{9}) + 32 i sin (\frac{11 π}{9})

Die Lösungen sind

z = 32 cos (\frac{π}{9}) + 32 i sin (\frac{π}{9}), z = 32 cos (\frac{7 π}{9}) + 32 i sin (\frac{7 π}{9}), z = 32 cos (\frac{13 π}{9}) + 32 i sin (\frac{13 π}{9}), z = 32 cos (\frac{π}{9}) - 32 i sin (\frac{π}{9}), z = 32 cos (\frac{5 π}{9}) + 32 i sin (\frac{5 π}{9}), z = 32 cos (\frac{11 π}{9}) + 32 i sin (\frac{11 π}{9})

Graph

Beliebte Beispiele

2 x^{3} + 9 x^{2} + 4 x = 0

x^{5} + 9 x = 0

x^{12} = 12

2 x^{3} - 4 x + 2 = 0

solvefor x,six^3=125

so l v e f or x, s i x^{3} = 125