Do I have to solve it iteratively?
Only up to quartic functions there exist generic analytic formulas for the roots. For higher degree polynomials it has been proven no such generic formulas exist and you have to use an algorithmic approach.
However, your specific polynomial can be rewritten as a quartic one:
0.0625 - 1.5*z + 7.5*z\\^2 - 14*z\\^3 + 9z\\^4 = 0
z = x\\^2
Solving for z gives me 0.5 (three times) and 0.055555. Taking their square roots to get x results in 0.70710678 and 0.23570226.