2x^4+3x^2+8=0

Lösung

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

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

Schritte zur Lösung

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

Schreibe die Gleichung um mit

u = x^{2}

und

u^{2} = x^{4}

2 u^{2} + 3 u + 8 = 0

Löse

2 u^{2} + 3 u + 8 = 0 : u = - \frac{3}{4} + i \frac{55}{4}, u = - \frac{3}{4} - i \frac{55}{4}

u = - \frac{3}{4} + i \frac{55}{4}, u = - \frac{3}{4} - i \frac{55}{4}

Setze

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

löse für

x

Löse

x^{2} = - \frac{3}{4} + i \frac{55}{4} : x = \frac{5}{2 2} + \frac{11}{2 2} i, x = - \frac{5}{2 2} - \frac{11}{2 2} i

Löse

x^{2} = - \frac{3}{4} - i \frac{55}{4} : x = \frac{5}{2 2} - \frac{11}{2 2} i, x = - \frac{5}{2 2} + \frac{11}{2 2} i

Die Lösungen sind

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

x^{3} - 5 x^{2} - 30 x - 54 = 0

4 x^{3} + 3 x^{2} + 100 x + 75 = 0

4 x^{3} = x

so l v e f or x, 2 y^{2} = x^{3}

x^{3} + 4 x^{2} - 11 x - 30 = 0