comment it s incomprehensible that a point of such importance symmetry inflection the solutions and critical points evaluation isn t mentioned within whole article moreover the circle at figure is unnecessary dislocated above s s y s omitting even a vertical line up to the center of the circle as done at figure of new approach by nickalls r w d see note again tschirnhaus transformation isn t only unneeded but also responsible for such a failure hence t s doesn t mean that an inflection point s q doesn t exist it seems to me reasonable the unknown t along with the variables p and q to be abandoned in favour of graphically grounded ones which are also easier for memorizing