\noindent{\bf Ans}:

Apply the following identity
$$
a^2-b^2 = (a+b)(a-b)
$$
that avoids the subtraction of two nearly identical numbers and gives
$$
f(x) = \frac{1.7x}{\sqrt{1+0.9x}+\sqrt{1-0.8x}} - 10^{-10}. 
\;\textrm{{\bf(7 pts)}}
$$
Then solve $f(x) = 0$ by any numerical method to find the solution
$$
x_* \approx 1.17647058823875 \times 10^{-10}.\;\textrm{{\bf(8 pts)}}
$$