# Is quintic equations have anything to do with their coefficients?

I was reading about quintic equations and this came up:

In the early nineteenth century, Paolo Ruffini (1765–1822) and Niels Henrik Abel (1802–1829) proved that no such general formulas utilizing the usual operations and root operations exist. This means that there will never be a simple formula that provides the solutions for every single a, b, c, d, e, and f in a quintic equation.

so, I am a little confused. isn't the goal of solving a quintic equation is to find X? what is the role of coefficients in here? can anyone explain?

• If you have no coefficients, do you even have a quintic to solve for $x$? – David K Jun 11 '19 at 16:50
• Not sure what you are asking. It is, of course, true that $\textit {some}$ quintics are solvable. $x^5$ for example, or $(x+1)(x+2)(x+3)(x+4)(x+5)$. The Theorem says that there isn't a "sensible" formula, akin to the quadratic formula, that solves all quintics regardless of the coefficients. – lulu Jun 11 '19 at 16:59

• @Daruissoli In other words, solving any problem of the form $ax^5+bx^4+cx^3+dx^2+ex+f=0$. – J.G. Jun 11 '19 at 20:41