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Picking genuine zeros of cubics in the Tschirnhaus method

In 1683, the German mathematician Ehrenfried Walther von Tschirnhaus introduced a polynomial transformation which, he claimed, would eliminate all intermediate terms in a polynomial equation of any degree, thereby reducing it to a binomial form from which roots can easily be extracted [1]. As mathematicians at that time were struggling to solve quintic equations in radicals with no sign of any success, the Tschirnhaus transformation gave them some hope, and in 1786, Bring was able to reduce the general quintic to the form x5 + ax + b = 0, even though he didn't succeed in his primary mission of solving it. It seems Bring's work got lost in the archives of University of Lund. Unaware of Bring's work, Jerrard (1859) also arrived at the same form of the quintic using a quartic Tschirnhaus transformation [2]. From the works of Abel (1826) and Galois (1832) it is now clear that the general polynomial equation of degree higher than four cannot be solved in radicals.