No solving for x x

Algebra Level 3

Given that x x is a complex number satisfying x + 1 x = 5 x + \dfrac{1}{x} = 5 , find the value of x 4 + 1 x 4 x^4 + \dfrac{1}{x^4} .


The answer is 527.

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Hobart Pao by Hobart Pao , 23, USA

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2 solutions

⇒ x 4 + 1 x 4 = y \Rightarrow x^4+\dfrac{1}{x^4}=y

[ ( x + 1 x ) 2 − 2 ] 2 − 2 = y \left [\left(x+\dfrac{1}{x}\right)^2-2 \right]^2-2=y

[ ( 5 ) 2 − 2 ] 2 − 2 = y \left [(5)^2-2\right ]^2-2=y

y = 527 y=527

⇒ x 4 + 1 x 4 = 527 \Rightarrow x^4+\dfrac{1}{x^4}=\boxed{527}

5 Helpful 1 Interesting 0 Brilliant 0 Confused
Arulx Z
Jan 25, 2016

x + 1 x = 5 ( x + 1 x ) 2 = 5 2 x 2 + 1 x 2 + 2 = 25 x 2 + 1 x 2 = 23 ( x 2 + 1 x 2 ) 2 = 23 2 x 4 + 1 x 4 + 2 = 529 x 4 + 1 x 4 = 527 x+\frac { 1 }{ x } =5\\ { \left( x+\frac { 1 }{ x } \right) }^{ 2 }={ 5 }^{ 2 }\\ { x }^{ 2 }+\frac { 1 }{ { x }^{ 2 } } +2=25\\ { x }^{ 2 }+\frac { 1 }{ { x }^{ 2 } } =23\\ { \left( { x }^{ 2 }+\frac { 1 }{ { x }^{ 2 } } \right) }^{ 2 }={ 23 }^{ 2 }\\ { x }^{ 4 }+\frac { 1 }{ { x }^{ 4 } } +2=529\\ { x }^{ 4 }+\frac { 1 }{ { x }^{ 4 } } =527

4 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

Simple standard approach.

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