Find All Real Solutions?

x - √(x² - x) = √(x)
x - √(x) = √(x² - x)
[x - √(x)]² = [√(x² - x)]²
x² - 2x√(x) + [√(x)]² = x² - x
x² - 2x√(x) + x = x² - x
2x√(x) - 2x = 0
x√(x) - x = 0
[√(x) - 1] x = 0
√x = 1 or x = 0 (rejected, for x > 0)
(√x)² = 1²
x = 1

A graph shows the real solutions are x = 0 or 1.

Click on the curve in the graph to highlight the solutions.

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If you subtract x and square both sides of the equation, you get
.. x^2 -x = x^2 -2x√x +x
Now subtract x^2+x and square again.
.. 4x^2 = 4x^3
.. x^2(x -1) = 0 . . . divide by 4, subtract the left side, factor
.. x = 0 or 1 . . . both answers check OK in the original equation