Algebraic Manipulation

Algebra Level 2

If x + 1 x = 7 x+\frac{1}{x}=7 , then what is x 2 + 1 x 2 x^2+\frac{1}{x^2} ?

51 47 49 343

Keshav Ramesh by Keshav Ramesh , 18, USA

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1 solution

Zee Ell
Feb 12, 2017

x + 1 x = 7 x + \frac {1}{x} = 7

( x + 1 x ) 2 = x 2 + 1 x 2 + 2 × x × 1 x = x 2 + 1 x 2 + 2 (x + \frac {1}{x} )^2 = x^2 + \frac {1}{x^2} + 2 × x × \frac {1}{x} = x^2 + \frac {1}{x^2} + 2

x 2 + 1 x 2 = ( x + 1 x ) 2 − 2 = 7 2 − 2 = 49 − 2 = 47 x^2 + \frac {1}{x^2} = ( x + \frac {1}{x} )^2 - 2 = 7^2 - 2 = 49 - 2 = \boxed {47}

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