We are learning a lot about the history of our famous mathematicians and this specific one is stumping me. They want us to solve a problem a specific way and I can't seem to figure out how to do it.
Would anyone be able to assist me in how to solve his quartic equation this specific way. A dying Italian mathematician revealed to you a secret formula for solving quartic equations of the form $x^4+ax^3+bx^2+c=0$. How would you use it to solve quartic equations of the form $u^4+du^2+fu+g=0$ ($u$ is the unknown here), where $g \not= 0$ ? In other words, what substitution $u=u(x)$ would you use to reduce equations of the form $u^4+du^2+fu+g=0$ to equations of the form $x^4+ax^3+bx^2+c=0$?
Hint: Because $g\ne 0$, $u=0$ is not a solution of our equation. Try the substitution $u=\frac{1}{x}$.