I have been studying on cubic equations for a while and see that the cubic equation needs to be in the form of so that we can find the roots easily. In order to obtain such an equation having no quadratic term for a cubic equation in the form of , value needs to be replaced with .
I realised that the second derivative of any cubic equation in the form of is , and when we equalize to , is equal to . This had me thinking about any possible relation between the quadratic term and the second derivative.
It is also sort of the same in quadratic functions in the form of . When we replace with , which is the first derivative of a quadratic function and the vertex point of it, we obtain the vertex form of the equation which does not have the linear term, .
Is there a relation between the quadratic term and the second derivative of a cubic equation? If the answer is yes, what is it and how is it observed on graphs, in equations?