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

Lösung

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

x = \frac{3 + 4 3}{2} + \frac{39}{2 3 + 4 3} i, x = - \frac{3 + 4 3}{2} - \frac{39}{2 3 + 4 3} i, x = \frac{3 + 4 3}{2} - \frac{39}{2 3 + 4 3} i, x = - \frac{3 + 4 3}{2} + \frac{39}{2 3 + 4 3} i

Schritte zur Lösung

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

Schreibe die Gleichung um mit

u = x^{2}

und

u^{2} = x^{4}

u^{2} - 3 u + 12 = 0

Löse

u^{2} - 3 u + 12 = 0 : u = \frac{3}{2} + i \frac{39}{2}, u = \frac{3}{2} - i \frac{39}{2}

u = \frac{3}{2} + i \frac{39}{2}, u = \frac{3}{2} - i \frac{39}{2}

Setze

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

löse für

x

Löse

x^{2} = \frac{3}{2} + i \frac{39}{2} : x = \frac{3 + 4 3}{2} + \frac{39}{2 3 + 4 3} i, x = - \frac{3 + 4 3}{2} - \frac{39}{2 3 + 4 3} i

Löse

x^{2} = \frac{3}{2} - i \frac{39}{2} : x = \frac{3 + 4 3}{2} - \frac{39}{2 3 + 4 3} i, x = - \frac{3 + 4 3}{2} + \frac{39}{2 3 + 4 3} i

Die Lösungen sind

x = \frac{3 + 4 3}{2} + \frac{39}{2 3 + 4 3} i, x = - \frac{3 + 4 3}{2} - \frac{39}{2 3 + 4 3} i, x = \frac{3 + 4 3}{2} - \frac{39}{2 3 + 4 3} i, x = - \frac{3 + 4 3}{2} + \frac{39}{2 3 + 4 3} i

- 4 x^{4} + 5 x^{3} - x^{2} = 0

6 x^{5} - 54 x = 0

x^{3} - 2 x^{2} - 14 x + 28 = 0

so l v e f or x, (2 x + 1)^{4} - 80 = 1

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