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Beliebt Algebra >

25x^4+15x^2+5=0

Lösung

25 x^{4} + 15 x^{2} + 5 = 0

Lösung

x = \frac{- 3 + 2 5}{2 5} + \frac{11}{2 5 - 3 + 2 5} i, x = - \frac{- 3 + 2 5}{2 5} - \frac{11}{2 5 - 3 + 2 5} i, x = \frac{- 3 + 2 5}{2 5} - \frac{11}{2 5 - 3 + 2 5} i, x = - \frac{- 3 + 2 5}{2 5} + \frac{11}{2 5 - 3 + 2 5} i

Schritte zur Lösung

25 x^{4} + 15 x^{2} + 5 = 0

Schreibe die Gleichung um mit

u = x^{2}

und

u^{2} = x^{4}

25 u^{2} + 15 u + 5 = 0

Löse

25 u^{2} + 15 u + 5 = 0 : u = - \frac{3}{10} + i \frac{11}{10}, u = - \frac{3}{10} - i \frac{11}{10}

u = - \frac{3}{10} + i \frac{11}{10}, u = - \frac{3}{10} - i \frac{11}{10}

Setze

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

löse für

x

Löse

x^{2} = - \frac{3}{10} + i \frac{11}{10} : x = \frac{- 3 + 2 5}{2 5} + \frac{11}{2 5 - 3 + 2 5} i, x = - \frac{- 3 + 2 5}{2 5} - \frac{11}{2 5 - 3 + 2 5} i

Löse

x^{2} = - \frac{3}{10} - i \frac{11}{10} : x = \frac{- 3 + 2 5}{2 5} - \frac{11}{2 5 - 3 + 2 5} i, x = - \frac{- 3 + 2 5}{2 5} + \frac{11}{2 5 - 3 + 2 5} i

Die Lösungen sind

x = \frac{- 3 + 2 5}{2 5} + \frac{11}{2 5 - 3 + 2 5} i, x = - \frac{- 3 + 2 5}{2 5} - \frac{11}{2 5 - 3 + 2 5} i, x = \frac{- 3 + 2 5}{2 5} - \frac{11}{2 5 - 3 + 2 5} i, x = - \frac{- 3 + 2 5}{2 5} + \frac{11}{2 5 - 3 + 2 5} i

Graph

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x^3-5x^2+25x-125=0

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