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Beliebt Algebra >

(x-2)^5=8(x-2)^3

Lösung

(x - 2)^{5} = 8 (x - 2)^{3}

Lösung

x = 2, x = 2 (1 + 2), x = 2 (1 - 2)

+1

Dezimale

x = 2, x = 4.82842 \dots, x = - 0.82842 \dots

Schritte zur Lösung

(x - 2)^{5} = 8 (x - 2)^{3}

Schreibe

(x - 2)^{5}

um:

x^{5} - 10 x^{4} + 40 x^{3} - 80 x^{2} + 80 x - 32

Schreibe

8 (x - 2)^{3}

um:

8 x^{3} - 48 x^{2} + 96 x - 64

x^{5} - 10 x^{4} + 40 x^{3} - 80 x^{2} + 80 x - 32 = 8 x^{3} - 48 x^{2} + 96 x - 64

Subtrahiere

8 x^{3} - 48 x^{2} + 96 x - 64

von beiden Seiten

x^{5} - 10 x^{4} + 40 x^{3} - 80 x^{2} + 80 x - 32 - (8 x^{3} - 48 x^{2} + 96 x - 64) = 8 x^{3} - 48 x^{2} + 96 x - 64 - (8 x^{3} - 48 x^{2} + 96 x - 64)

Vereinfache

x^{5} - 10 x^{4} + 32 x^{3} - 32 x^{2} - 16 x + 32 = 0

Faktorisiere

x^{5} - 10 x^{4} + 32 x^{3} - 32 x^{2} - 16 x + 32 : (x - 2)^{3} (x^{2} - 4 x - 4)

(x - 2)^{3} (x^{2} - 4 x - 4) = 0

Anwendung des Nullfaktorprinzips: Wenn

ab = 0

dann

a = 0

oder

b = 0

x - 2 = 0 or x^{2} - 4 x - 4 = 0

Löse

x - 2 = 0 : x = 2

Löse

x^{2} - 4 x - 4 = 0 : x = 2 (1 + 2), x = 2 (1 - 2)

Die Lösungen sind

x = 2, x = 2 (1 + 2), x = 2 (1 - 2)

Graph

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2 x^{3} + 3 x^{2} - 5 x + 74 = 3 x^{2} + 8 x - 6

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