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An equation of the form EQN:ax^5+bx^4+cx^3+dx^2+ex+f=0

It is well know that the quadratic equation has a closed form solution.

It is well known that the quadratic equation has a closed form solution.

It is less well known that the cubic equation and quartic equation also

both have closed form solutions, although they are significantly more

complex.

For centuries a general solution for the quintic (and higher) equation

was sought, but in 1824 Abel proved that such a formula (using radicals)

is impossible, and this was generalised and extended by Galois's work of

1832 or so.