The known algorithms approximate the roots of a univariate polynomial in nearly optimal arithmetic and Boolean time. They are, however, are quite involved and require a high precision of computing when the degree of the input polynomial is large, which causes numerical stability problems. We observe that these difficulties do not appear at the initial stage of the algorithms, and in our present paper we extend this stage, analyze it, and avoid the cited problems, still achieving the solution at a nearly optimal estimated complexity, provided that some mild initial isolation of the roots of the input polynomial is ensured. The resulting algorithms promise to be of practical value for root-finding and can be extended to the problem of polynomial factorization, which has also interest on its own.