What is the solution of the equation (x^3 - x)^1/2 + (2x - 1)^1/2 = (x^3 + x - 1)^1/2?

Here we need to solve the equation (x^3 - x)^1/2 + (2x -
1)^1/2 = (x^3 + x - 1)^1/2 for x.

We go about it in the
following way. First eliminate the power 1/2 that has been applied to all the terms.
This can be done by taking the square of both the
sides.

=> [(x^3 - x) ^ (1/2) + (2x - 1) ^ (1/2)] ^2
= [(x^3 + x - 1) ^ (1/2)] ^2

=> (x^3 - x) + (2x - 1)
+2*(x^3 - x) ^ (1/2) *(2x - 1) ^ (1/2) = (x^3 + x -
1)

=> x^3 – x + 2x – 1 + 2*(2x^4 – x^3 – 2x^2 + x) ^
(1/2) = x^3 + x -1

cancel the common
terms

=> 2*(2x^4 – x^3 – 2x^2 + x) ^ (1/2) =
0

=> (2x^4 – x^3 – 2x^2 + x) ^ (1/2) =
0

square both the
sides

=> 2x^4 – x^3 – 2x^2 + x =
0

=> x (2x^3 – x^2 – 2x + 1) =
0

=> x [x^2(2x-1) -1(2x-1)] =
0

=> x(x-1) (x+1) (2x-1)
=0

=> x = 0 or x-1=0 or x+1=0 or
2x-1=0

=> x = 0 or x = 1 or x = -1 or x =
1/2

Therefore the solutions for x are 0, 1,
-1 and 1/2.