Change coordinates by substituting $y=z+ \lambda/z$. What is this?

Asked Mar 30 '20 at 22:16

Active Mar 30 '20 at 22:28

Viewed 77 times

So I had to find the roots of this equation:

$$x^3−2x^2−x+6=0$$ but could not figure out how. Since I couldn't figure it out I put it on wolfram alpha on my phone. It used two methods that I had never heard of before. The two things it did was "Eliminating the quadratic term by substituting $y=x−\frac{2}{3}$ " and then "Change coordinates by substituting $y=z+\frac{\lambda}{z}$ , where λ is a constant value that will be determined later".

My question is about that second method they used. What exactly is that method doing and why is it useful? In what cases would it be helpful to do that?

Sorry if it sounds like I am asking for a whole lesson. If possible, maybe someone could just tell me what that method is called so then I could go and learn about it.

edited Mar 30 '20 at 22:28

xxxxxxxxx

13,302

asked Mar 30 '20 at 22:16

Angel Borge

3

See Cubic equation, in particular the sections on "Depressed cubic" and "Vieta's substitution". – Robert Israel Mar 30 '20 at 22:24
Those aren't two methods, Angel. They are two steps in a single method; first you eliminate the quadratic term, then you change the coordinates. If you follow the second step through, you can't help but see what it does, and why it's useful; it changes your cubic in $y$ to a quadratic in $z^3$, which is great because you know how to solve quadratics. – Gerry Myerson Mar 31 '20 at 00:11
Ah I see now thank you! Does this method work for most cubics or only limited cases of them? – Angel Borge Mar 31 '20 at 02:33
All cubics. Did you follow the link Robert Israel gave? – Gerry Myerson Mar 31 '20 at 11:24
I say, did you follow the link Robert Israel gave? – Gerry Myerson Apr 01 '20 at 23:25
1

Hi! Yes I did I am going to search up more practice with it now that I know what it is! – Angel Borge Apr 02 '20 at 02:25

Change coordinates by substituting $y=z+ \lambda/z$. What is this?

0 Answers0