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Beliebt Algebra >

(1+x)^{15}=50x+1

Lösung

(1 + x)^{15} = 50 x + 1

Lösung

x = 0, x \approx 0.15603 \dots, x \approx - 2.37421 \dots

Schritte zur Lösung

(1 + x)^{15} = 50 x + 1

Schreibe

(1 + x)^{15}

um:

1 + 15 x + 105 x^{2} + 455 x^{3} + 1365 x^{4} + 3003 x^{5} + 5005 x^{6} + 6435 x^{7} + 6435 x^{8} + 5005 x^{9} + 3003 x^{10} + 1365 x^{11} + 455 x^{12} + 105 x^{13} + 15 x^{14} + x^{15}

1 + 15 x + 105 x^{2} + 455 x^{3} + 1365 x^{4} + 3003 x^{5} + 5005 x^{6} + 6435 x^{7} + 6435 x^{8} + 5005 x^{9} + 3003 x^{10} + 1365 x^{11} + 455 x^{12} + 105 x^{13} + 15 x^{14} + x^{15} = 50 x + 1

Verschiebe

1

auf die rechte Seite

15 x + 105 x^{2} + 455 x^{3} + 1365 x^{4} + 3003 x^{5} + 5005 x^{6} + 6435 x^{7} + 6435 x^{8} + 5005 x^{9} + 3003 x^{10} + 1365 x^{11} + 455 x^{12} + 105 x^{13} + 15 x^{14} + x^{15} = 50 x

Verschiebe

50 x

auf die linke Seite

x^{15} + 15 x^{14} + 105 x^{13} + 455 x^{12} + 1365 x^{11} + 3003 x^{10} + 5005 x^{9} + 6435 x^{8} + 6435 x^{7} + 5005 x^{6} + 3003 x^{5} + 1365 x^{4} + 455 x^{3} + 105 x^{2} - 35 x = 0

Faktorisiere

x^{15} + 15 x^{14} + 105 x^{13} + 455 x^{12} + 1365 x^{11} + 3003 x^{10} + 5005 x^{9} + 6435 x^{8} + 6435 x^{7} + 5005 x^{6} + 3003 x^{5} + 1365 x^{4} + 455 x^{3} + 105 x^{2} - 35 x : x (x^{14} + 15 x^{13} + 105 x^{12} + 455 x^{11} + 1365 x^{10} + 3003 x^{9} + 5005 x^{8} + 6435 x^{7} + 6435 x^{6} + 5005 x^{5} + 3003 x^{4} + 1365 x^{3} + 455 x^{2} + 105 x - 35)

x (x^{14} + 15 x^{13} + 105 x^{12} + 455 x^{11} + 1365 x^{10} + 3003 x^{9} + 5005 x^{8} + 6435 x^{7} + 6435 x^{6} + 5005 x^{5} + 3003 x^{4} + 1365 x^{3} + 455 x^{2} + 105 x - 35) = 0

Anwendung des Nullfaktorprinzips: Wenn

ab = 0

dann

a = 0

oder

b = 0

x = 0 or x^{14} + 15 x^{13} + 105 x^{12} + 455 x^{11} + 1365 x^{10} + 3003 x^{9} + 5005 x^{8} + 6435 x^{7} + 6435 x^{6} + 5005 x^{5} + 3003 x^{4} + 1365 x^{3} + 455 x^{2} + 105 x - 35 = 0

Löse

x^{14} + 15 x^{13} + 105 x^{12} + 455 x^{11} + 1365 x^{10} + 3003 x^{9} + 5005 x^{8} + 6435 x^{7} + 6435 x^{6} + 5005 x^{5} + 3003 x^{4} + 1365 x^{3} + 455 x^{2} + 105 x - 35 = 0 : x \approx 0.15603 \dots, x \approx - 2.37421 \dots

Die Lösungen sind

x = 0, x \approx 0.15603 \dots, x \approx - 2.37421 \dots

Graph

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