# Parameters for Plotting a Quartic

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The general quartic can be brought into the reduced form

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Contributed by: Thomas Mueller and R.W.D. Nickalls (March 2011)

Open content licensed under CC BY-NC-SA

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The four roots of the reduced quartic equation can be expressed in terms of just three parameters, say , , , where , , are the roots of the resolvent cubic equation , known as Euler's cubic. Note that are the six roots of the resolvent sextic Thus, the key to solving the quartic is to first solve the resolvent cubic

For a more detailed discussion see R. W. D. Nickalls, "The Quartic Equation: Invariants and Euler's Solution Revealed," *The Mathematical Gazette,* 94, 2009 pp. 66–75.

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