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Level 2

If fractional part of x 2 x^2 and 1 x \dfrac{1}{x} for some x ∈ ( 2 , 3 ) x\in (\sqrt 2, \sqrt 3) are equal then what is the sum of all the distinct possible values of x 4 − 3 x x^{4} -\dfrac{3}{x} ?

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Ruchit Chudasama by Ruchit Chudasama , 19, India

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1 solution

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Jul 17, 2018

x 2 ∈ ( 2 , 3 ) , 1 x ∈ ( 0 , 1 ) x^2\in ( 2, 3),\dfrac1x\in (0, 1) ,so x 2 − 1 x = 2 , x = 1 + 5 2 , x 4 − 3 x = 5 x^2-\dfrac1x=2,x=\dfrac{1+\sqrt5}2,x^4-\dfrac3x=5

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