Parameters for Plotting a Quartic

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The general quartic can be brought into the reduced form
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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