Is there a general formula the solutions of a polynomial equation of the form $Ax^n + Bx^{n-1} + C = 0$?

Question

Is there a general formula for the solutions of a polynomial equation of the form $$Ax^n + Bx^{n-1} + C = 0,$$ where $A$, $B$, $C$, and $n$ are constants?

Answer 1

The Galois group of the splitting field of, for example, $x^5 - x^4 + 1$, is isomorphic to $S_5$, which is not solvable, and hence the roots of that polynomial are not solvable in radicals. Hence, there is no general formula for the roots of a real polynomial of the form $A x^n + B x^{n - 1} + C$ in terms of radicals. (See the Wikipedia article on the Abel-Ruffini Theorem for more.)