This chart shows how to solve every quartic polynomial.

Notice it also shows how to solve cubics and quadratics, but it can’t be improved to quintics (polynomials of degree five) or higher, because only polynomials with degree less than five can be solved algebraically in general: this is the Abel-Ruffini theorem. Some specific quintics can be solved, but the method is far more tedious. In 2004 Daniel Lazard wrote out a three-page formula for the roots of a general solvable quintic.

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