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Beliebt Algebra >

16z^4-4z^2+1=0

Lösung

16 z^{4} - 4 z^{2} + 1 = 0

Lösung

z = \frac{3}{4} + \frac{1}{4} i, z = - \frac{3}{4} - \frac{1}{4} i, z = \frac{3}{4} - \frac{1}{4} i, z = - \frac{3}{4} + \frac{1}{4} i

Schritte zur Lösung

16 z^{4} - 4 z^{2} + 1 = 0

Schreibe die Gleichung um mit

u = z^{2}

und

u^{2} = z^{4}

16 u^{2} - 4 u + 1 = 0

Löse

16 u^{2} - 4 u + 1 = 0 : u = \frac{1}{8} + i \frac{3}{8}, u = \frac{1}{8} - i \frac{3}{8}

u = \frac{1}{8} + i \frac{3}{8}, u = \frac{1}{8} - i \frac{3}{8}

Setze

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

löse für

z

Löse

z^{2} = \frac{1}{8} + i \frac{3}{8} : z = \frac{3}{4} + \frac{1}{4} i, z = - \frac{3}{4} - \frac{1}{4} i

Löse

z^{2} = \frac{1}{8} - i \frac{3}{8} : z = \frac{3}{4} - \frac{1}{4} i, z = - \frac{3}{4} + \frac{1}{4} i

Die Lösungen sind

z = \frac{3}{4} + \frac{1}{4} i, z = - \frac{3}{4} - \frac{1}{4} i, z = \frac{3}{4} - \frac{1}{4} i, z = - \frac{3}{4} + \frac{1}{4} i

Graph

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