Number of real roots of the equation $\sqrt{x}+\sqrt{x-\sqrt{1-x}}=1$ is

The domain of the equation requires x to be between 0 and 1 inclusive. Squaring both sides yields 2x minus 1 equals 2 times the square root of the quantity x times the quantity 1 minus x. Squaring again and rearranging terms gives the quadratic 8x squared minus 8x plus 1 equals 0. Solving this with the quadratic formula gives two valid solutions, x equals 1/2 plus the square root of 2 over 4 and x equals 1/2 minus the square root of 2 over 4. The result is 2.

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