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x^3+1/x^3 = 36 then x +1/x =?

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stjmkt1 | Teacher | eNotes Newbie

Posted November 24, 2011 at 12:16 PM via web

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x^3+1/x^3 = 36 then x +1/x =?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted November 24, 2011 at 5:04 PM (Answer #1)

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Use the formula `a^3 + b^3=(a+b)(a^2-ab+b^2)` `x^3+1/x^3=(x+1/x)(x^2-x*(1/x)+1/x^2)`

`` `x^2+1/x^2 = (x+1/x)^2- 2 =gt`

`=gtx^3+1/x^3=(x+1/x)((x-1/x)^2- 2 - 1)` `x^3+1/x^3=(x+1/x)((x+1/x)^2 -3)`

`` `36=(x+1/x)^3 - 3(x+1/x)`

Let `x+1/x=y`

`36=y^3-3y =gt y^3-3y-36=0`

`` `y(y^2-3)=36`

`y^2-3=6 =gt y^2=9 =gt y1=3 and y2=-3`

But 3*6=18`!=` 36 =>`y!=+-3` => `x+1/x!=+-3`

`y^2-3=9=gty^2=12`  =>`y_(1,2)=+-2sqrt3`

Notice that `2sqrt3*9=18sqrt3!=36 =gt x+1/x!=+-2sqrt3`

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