10x^4-3x^2+1=0

Lösung

10 x^{4} - 3 x^{2} + 1 = 0

x = \frac{3 + 2 10}{2 10} + \frac{31}{2 10 3 + 2 10} i, x = - \frac{3 + 2 10}{2 10} - \frac{31}{2 10 3 + 2 10} i, x = \frac{3 + 2 10}{2 10} - \frac{31}{2 10 3 + 2 10} i, x = - \frac{3 + 2 10}{2 10} + \frac{31}{2 10 3 + 2 10} i

Schritte zur Lösung

10 x^{4} - 3 x^{2} + 1 = 0

Schreibe die Gleichung um mit

u = x^{2}

und

u^{2} = x^{4}

10 u^{2} - 3 u + 1 = 0

Löse

10 u^{2} - 3 u + 1 = 0 : u = \frac{3}{20} + i \frac{31}{20}, u = \frac{3}{20} - i \frac{31}{20}

u = \frac{3}{20} + i \frac{31}{20}, u = \frac{3}{20} - i \frac{31}{20}

Setze

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

löse für

x

Löse

x^{2} = \frac{3}{20} + i \frac{31}{20} : x = \frac{3 + 2 10}{2 10} + \frac{31}{2 10 3 + 2 10} i, x = - \frac{3 + 2 10}{2 10} - \frac{31}{2 10 3 + 2 10} i

Löse

x^{2} = \frac{3}{20} - i \frac{31}{20} : x = \frac{3 + 2 10}{2 10} - \frac{31}{2 10 3 + 2 10} i, x = - \frac{3 + 2 10}{2 10} + \frac{31}{2 10 3 + 2 10} i

Die Lösungen sind

x = \frac{3 + 2 10}{2 10} + \frac{31}{2 10 3 + 2 10} i, x = - \frac{3 + 2 10}{2 10} - \frac{31}{2 10 3 + 2 10} i, x = \frac{3 + 2 10}{2 10} - \frac{31}{2 10 3 + 2 10} i, x = - \frac{3 + 2 10}{2 10} + \frac{31}{2 10 3 + 2 10} i

x^{3} + 7 x^{2} - 4 x - 28 = 0

x^{4} - 2 x^{3} + x^{2} - 8 x - 12 = 0

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

12 x^{3} - 20 x^{2} + 11 x - 2 = 0

0 = 3 x^{3} - x^{2} - 12 x + 4