If x^3 + 1/x^3 = 10, then find x + 1/x.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to write an alternate form of the given expression `x^3 + 1/x^3 = 10`  such that:

`x^3 + 1/x^3 = 10 => x^6 + 1 = 10x^3`

Using the following substitution yields:

`x^3 = t => x^6 = t^2`

Changing the variable yields:

`t^2 - 10t + 1 = 0`

You should use quadratic formula such that:

`t_(1,2) = (10+-sqrt(100-4))/2`

`t_(1,2) = (10+-4sqrt(6)/2 => t_(1,2) = 5+-2sqrt6`

You should substitute `x^3`  for `t`  such that:

`x^3 = 5+-2sqrt6 => x = root(3)(5+-2sqrt6)`

You may evaluate `x + 1/x`  such that:

`x + 1/x =root(3)(5+-2sqrt6) + 1/(root(3)(5+-2sqrt6))`

Hence, evaluating the expression `x + 1/x`  yields `x + 1/x = root(3)(5+-2sqrt6) + 1/(root(3)(5+-2sqrt6)).`

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