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Beliebt Algebra >

x^{10}=-x

Lösung

x^{10} = - x

Lösung

x = 0, x = - 1, x = \frac{1}{2} + i \frac{3}{2}, x = \frac{1}{2} - i \frac{3}{2}, x = cos (\frac{π}{9}) + i sin (\frac{π}{9}), x = cos (\frac{7 π}{9}) + i sin (\frac{7 π}{9}), x = cos (\frac{13 π}{9}) + i sin (\frac{13 π}{9}), x = cos (\frac{π}{9}) - i sin (\frac{π}{9}), x = cos (\frac{5 π}{9}) + i sin (\frac{5 π}{9}), x = cos (\frac{11 π}{9}) + i sin (\frac{11 π}{9})

Schritte zur Lösung

x^{10} = - x

Verschiebe

x

auf die linke Seite

x^{10} + x = 0

Faktorisiere

x^{10} + x : x (x + 1) (x^{2} - x + 1) (x^{6} - x^{3} + 1)

x (x + 1) (x^{2} - x + 1) (x^{6} - x^{3} + 1) = 0

Anwendung des Nullfaktorprinzips: Wenn

ab = 0

dann

a = 0

oder

b = 0

x = 0 or x + 1 = 0 or x^{2} - x + 1 = 0 or x^{6} - x^{3} + 1 = 0

Löse

x + 1 = 0 : x = - 1

Löse

x^{2} - x + 1 = 0 : x = \frac{1}{2} + i \frac{3}{2}, x = \frac{1}{2} - i \frac{3}{2}

Löse

x^{6} - x^{3} + 1 = 0 : x = cos (\frac{π}{9}) + i sin (\frac{π}{9}), x = cos (\frac{7 π}{9}) + i sin (\frac{7 π}{9}), x = cos (\frac{13 π}{9}) + i sin (\frac{13 π}{9}), x = cos (\frac{π}{9}) - i sin (\frac{π}{9}), x = cos (\frac{5 π}{9}) + i sin (\frac{5 π}{9}), x = cos (\frac{11 π}{9}) + i sin (\frac{11 π}{9})

Die Lösungen sind

x = 0, x = - 1, x = \frac{1}{2} + i \frac{3}{2}, x = \frac{1}{2} - i \frac{3}{2}, x = cos (\frac{π}{9}) + i sin (\frac{π}{9}), x = cos (\frac{7 π}{9}) + i sin (\frac{7 π}{9}), x = cos (\frac{13 π}{9}) + i sin (\frac{13 π}{9}), x = cos (\frac{π}{9}) - i sin (\frac{π}{9}), x = cos (\frac{5 π}{9}) + i sin (\frac{5 π}{9}), x = cos (\frac{11 π}{9}) + i sin (\frac{11 π}{9})

Graph

Beliebte Beispiele

-x^3+9x^2-24x+20=0

- x^{3} + 9 x^{2} - 24 x + 20 = 0

(x^2-8)^2-2(x^2-8)+1=0

(x^{2} - 8)^{2} - 2 (x^{2} - 8) + 1 = 0

- 4 x^{5} = 256 x^{2}

8 = a^{3}

0=2x^3+8x^2+14x+24

0 = 2 x^{3} + 8 x^{2} + 14 x + 24