I need to determine the roots of the following function analytically:

$$f(x) = 1 - x - x \cdot e^{-2x}$$

This is my try on it:

$0 = 1 - x - x\*e^{-2x}$

$-1 = - x - x\*e^{-2x}$

$-1 + x = - x\*e^{-2x}$

$\frac{1 - x}{x} = e^{-2x}$

$\ln(1 - x)-\ln(x) = -2x$

$-\frac{1}{2}\ln(1 - x)+\frac{1}{2}\ln(x) = x\quad $

It would be great if someone could give me a hand! Thank you!