An easy Algebra challenge.

Algebra Level 2

If x x is a real number such that x 3 + 1 x 3 = 110 , x^{3} + \frac{1}{ x^{3}} = 110, Then find the value of, x + 1 x x + \frac{1}{x}


The answer is 5.

Yajur Phullera by Yajur Phullera , 19, India

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2 solutions

Raj Rajput
Sep 3, 2015

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Nice solution as usual Calligraphist Rajput.

Kushagra Sahni - 5 years, 9 months ago

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THANKS @Kushagra Sahni :) :)

RAJ RAJPUT - 5 years, 9 months ago
Yajur Phullera
Sep 3, 2015

Given that, x 3 + 1 x 3 = 110. x^{3} + \frac{1}{x^{3}}=110. Let x + 1 x = a . x + \frac{1}{x} =a. ( x + 1 x ) 3 = x 3 + 1 x 3 + 3 x 1 x ( x + 1 x ) (x + \frac{1}{x})^{3} = x^{3} + \frac{1}{x^{3}} + 3x\frac{1}{x}(x + \frac{1}{x}) a 3 = 110 + 3 ( x + 1 x ) a^{3} = 110 + 3(x + \frac{1}{x}) . a 3 = 110 + 3 a a^{3} = 110 + 3a a 3 − 3 a − 110 = 0 a^{3} -3a-110=0 Now by factor theorem, We get (a-5) only real factor ,of this equation. Therefore, a − 5 = 0 a-5= 0 a = 5 a=5 Hence x + 1 x = 5 x+ \frac{1}{x}=5

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