Why is x=1 and not 4 on this hard problem?

GRE Quant Problem Solving
1

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Hi I am quite puzzled about this solution to hard quant problem. I simplified it differently:

2 = √x + x(2-√x)
2 = √x + 2x -x√x
2-2x = √x-x√x

2(1-x) = √x(1-x)
2 = √x
4 = x

What am I doing wrong and what’s the error in my logic? As you can see having x=4 makes it hard for me to make sense of the answer choices.

I am finding with other hard math problems I seem to go on to more the difficult routes in my attempts. Any tips to improve or “see” how to approach hard problems would be appreciated.

2

If you decide to divide both sides by (1 - x), your subsequent equation operates on the premise that x \neq 1. Thus, it makes sense to check whether x = 1 is also a solution by plugging it into the original equation.

3

Ok.So make sure I understand correctly, the only 2 ways to have a reals solution is when x=1 and 4. Everything else is is not a real solution.

4

Those are indeed the two solutions, yes.

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