Understanding how the proof introduced a new coeffecient

May 2012
39
1
USA
Understanding the proof of quartic formula

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.
 

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Plato

MHF Helper
Aug 2006
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Re: Understanding the proof of quartic formula

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. What is the substitution to a previous line. Thank you for your help.
Do not post questions as attachments.
Are you so lazy as to not be able to type out the question?
Also you you can use LaTeX code on this site.
If you really want help, do one of those two options.
 
May 2012
39
1
USA
No. I am not lazy, but why are you attacking me. I have hard time use LaTex. The writing whole proof will be a boondoggle of time.
 

Plato

MHF Helper
Aug 2006
22,506
8,663
No. I am not lazy, but why are you attacking me. I have hard time use LaTex. The writing whole proof will be a boondoggle of time.
Why do you ask us to download an unknown file to our computer?
Type your question. It cannot be that hard. Otherwise you are lazy.
 
May 2012
39
1
USA
i will give the website, then. will it be fine then?
 

Prove It

MHF Helper
Aug 2008
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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?
 
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Prove It

MHF Helper
Aug 2008
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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.
 
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