I don't understand the latter part of the first sentence, ' only of for some ...' .
Anyway, the iteration given will not converge to either root of the quadratic, (unless the first iterate happens to be a root, if that can be called convergence).
Suppose that a root of the quadratic is and that the error in the n'th iterate is so that and similarly
Substitute those into the iterative formula and simplify. For convergence, has to be less than