Hello. I have some difficulties understanding a proof of solving quartic equation by using quartic formula. On page 2 of the proof, the author had introduced a new coefficient. Is there substitution from any line? What is the substitution? Thank you for your help.
There's no need for anyone to get upset. The point Plato is making is that most members here do not like to download unknown files to their computer, because there is the risk of downloading viruses. On the other hand, the OP has stated that this is a long proof, and typing it out is long and tedious, especially not being familiar with LaTeX.
On the subject of the original question, which is the "new coefficient" you do not understand?
I believe you need to read the "Solving Cubics" PDF in order to understand what is happening in that step.
Edit: No you don't. What it is saying is that with the lines talking about solving for z, all it's saying is that in order for the RHS of your equation involving y to be a perfect square, then the discriminant must be 0, which in turn creates a cubic equation in z. Since all cubics have at least one solution, then there is definitely a value of z which satisfies the RHS of your equation involving y being turned into a perfect square. Therefore, it is reasonable that you can write the RHS as (Sy + T)^2, where S and T are some constants.