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Solve for x  sqrt(x+3)-x^(1/3)=1

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pavelpimen | Student, Undergraduate | Honors

Posted December 1, 2011 at 1:03 AM via web

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Solve for x

 

sqrt(x+3)-x^(1/3)=1

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted December 1, 2011 at 1:41 AM (Answer #1)

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You need to solve the equation  `sqrt (x+3)`  - `root(3)(x)`  = 1

Notice that the one term of equation is a square root and the other term is a cube root.

Put `sqrt(x+3)`  = y and `root(3)(x)`  = z

Write the equation: y - z = 1 and`y^2 - z^3`  = 3

`y = z + 1 =gt (z+1)^2 - z^3 = 3 =gt z^2 + 2z + 1 - z^3- 3`  = 0

`z^2 + 2z - z^3 - 2 ` = 0

`(z^2 - 2) - z(z^2 - 2)`  = 0 =>`(z^2 - 2)(1 - z)`  = 0

`z^2 - 2 = 0 =gt z_(1,2) = +-sqrt2`

`1 - z = 0 =gt z = 1`

Take `z = 1 =gt y = 1+1 = 2 =gt x = z^3 = 1`

Take `z = sqrt 2 =gt y = sqrt 2 + 1 =gt x = 2sqrt2`  

Take `z = -sqrt 2 =gt z = 1 - sqrt 2 =gt x = -2sqrt 2`

The values of x will be: `x = -2sqrt 2 ; x = 1 ; x = 2 sqrt2` .

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