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Beliebt Algebra >

(x-3)^7=64(x-3)^4

Lösung

(x - 3)^{7} = 64 (x - 3)^{4}

Lösung

x = 3, x = 7, x = 1 + 23 i, x = 1 - 23 i

Schritte zur Lösung

(x - 3)^{7} = 64 (x - 3)^{4}

Schreibe

(x - 3)^{7}

um:

x^{7} - 21 x^{6} + 189 x^{5} - 945 x^{4} + 2835 x^{3} - 5103 x^{2} + 5103 x - 2187

Schreibe

64 (x - 3)^{4}

um:

64 x^{4} - 768 x^{3} + 3456 x^{2} - 6912 x + 5184

x^{7} - 21 x^{6} + 189 x^{5} - 945 x^{4} + 2835 x^{3} - 5103 x^{2} + 5103 x - 2187 = 64 x^{4} - 768 x^{3} + 3456 x^{2} - 6912 x + 5184

Subtrahiere

64 x^{4} - 768 x^{3} + 3456 x^{2} - 6912 x + 5184

von beiden Seiten

x^{7} - 21 x^{6} + 189 x^{5} - 945 x^{4} + 2835 x^{3} - 5103 x^{2} + 5103 x - 2187 - (64 x^{4} - 768 x^{3} + 3456 x^{2} - 6912 x + 5184) = 64 x^{4} - 768 x^{3} + 3456 x^{2} - 6912 x + 5184 - (64 x^{4} - 768 x^{3} + 3456 x^{2} - 6912 x + 5184)

Vereinfache

x^{7} - 21 x^{6} + 189 x^{5} - 1009 x^{4} + 3603 x^{3} - 8559 x^{2} + 12015 x - 7371 = 0

Faktorisiere

x^{7} - 21 x^{6} + 189 x^{5} - 1009 x^{4} + 3603 x^{3} - 8559 x^{2} + 12015 x - 7371 : (x - 3)^{4} (x - 7) (x^{2} - 2 x + 13)

(x - 3)^{4} (x - 7) (x^{2} - 2 x + 13) = 0

Anwendung des Nullfaktorprinzips: Wenn

ab = 0

dann

a = 0

oder

b = 0

x - 3 = 0 or x - 7 = 0 or x^{2} - 2 x + 13 = 0

Löse

x - 3 = 0 : x = 3

Löse

x - 7 = 0 : x = 7

Löse

x^{2} - 2 x + 13 = 0 : x = 1 + 23 i, x = 1 - 23 i

Die Lösungen sind

x = 3, x = 7, x = 1 + 23 i, x = 1 - 23 i

Graph

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