Expand (1-x)^(1/2) * (1+2x)^(1/2)

(1-x)^(1/2) * (1+2x)^(1/2)

= 1 + (1/2) x - (9/8)x^2 + (9/16)^3 - (117/128)x^4 + (279/256)x^5 + ...

the range of validity for the expansion of (1-x)^(1/2) is |x|<1

the range of validity for the expansion of (1+2x)^(1/2) is |x|< 1/2

taking the intersection, the expansion of (1-x)^(1/2) * (1+2x)^(1/2) is valid for |x|< 1/2.

Here come my questions,

(1-x)^(1/2) * (1+2x)^(1/2)

=(1+x-2x^2)^(1/2)

if I expand using Binomial theorem, the result is exactly the same.

However, when I solve |x-2x^2|<1 for the range of validity,

I got -0.414<x<1 or 1<x<2.414 (rejected as 1-x should be positive)

So -0.414<x<1!!! which is different from the previous |x|<1/2

You have solved |x-2x^2|<1 incorrectly.
May I ask why is the range of validity different? I thought since I am expanding the same expression so the range of validity should be unique, right?

Pls enlighten me