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?
The role of coefficients is to specify which polynomial function is in the problem. There is no general solution in radicals analogous to, say, the quadratic formula; that's what the theorem is saying. (In the quadratic case, the formula writes roots in terms of coefficients.)