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Beliebt Algebra >

(x^2+1)+(x^2+1)(x^4+1)=0

Lösung

(x^{2} + 1) + (x^{2} + 1) (x^{4} + 1) = 0

Lösung

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

Schritte zur Lösung

(x^{2} + 1) + (x^{2} + 1) (x^{4} + 1) = 0

Schreibe

(x^{2} + 1) + (x^{2} + 1) (x^{4} + 1)

um:

x^{6} + x^{4} + 2 x^{2} + 2

x^{6} + x^{4} + 2 x^{2} + 2 = 0

Schreibe die Gleichung um mit

u = x^{2}, u^{2} = x^{4}

und

u^{3} = x^{6}

u^{3} + u^{2} + 2 u + 2 = 0

Löse

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

u = - 1, u = 2 i, u = - 2 i

Setze

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

löse für

x

Löse

x^{2} = - 1 : x = i, x = - i

Löse

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

Löse

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

Die Lösungen sind

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

Graph

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