What is the value of x in the equation (1+x)^(2/3)+(1-x)^(2/3) = 4(1-x^2)^(1/3)

Question

Answer choices are where Correct choice is C. 

(A) 5/(3)^(1/3)
(B) -5/(3)^(1/3)
(C) +- 5/3(3)^(1/2)
(D) +- 15/(3)^1/3

 

When I substitute 5/3(3)^(1/2) or - 5/3(3)^(1/2) in (1+x)^(2/3)+(1-x)^(2/3) -4(1-x^2)^(1/3) I get a zero.

Therefore the equation is correct but I am unable to simplify the equation to come to this answer.

Answer 1

An analytic approach is to write the right hand side as:  ( 1 + x)^1/3  (1-x )^1/3    

 

Then define Q as   Q =  [ (1 -x )/(1 + x) ]^1/3    

 

After some rearrangement  the equation becomes

 

1/Q  + Q = 4    This can be rearranged as a quadratic equation and the quadratic formula used to get

   

  Q = 2 + sqrt(3)     or 1/ ( 2 + sqrt(3))

 

The defining equation for Q  can be rearranged to get   x = ( 1- Q³) / ( 1 + Q³ )

 

From there it is just a matter of substituting for Q  in this last equation.

 

After some very messy algebraic simplification , your answer emerges.