We have to solve: sqrt(1 - sqrt (x^4 - x)) = x - 1

sqrt(1 - sqrt (x^4 - x)) = x - 1

square both the sides

=> (1 - sqrt (x^4 - x)) = (x - 1)^2

=> (1 - sqrt (x^4 - x)) = (x^2 + 1 - 2x)

=> -sqrt (x^4 - x) = x^2 - 2x

square both the sides again

=> x^4 - x = x^4 + 4x^2 - 4x^3

=> 4x^3 - 4x^2 - x = 0

=> x(4x^2 - 4x - 1) = 0

x1 = 0

x2 = 4/8 + sqrt (16 + 16)/8

=> 1/2 + 4*sqrt 2/8

=> 1/2 + 1/sqrt 2

The other root of the quadratic equation is not a valid solution.

**The values are x = 0 and x = 1/2 + 1/sqrt 2**