A Complete Review of the General Quartic Equation with Real Coefficients and Multiple Roots

Mauricio Chávez-Pichardo, Miguel A. Martínez-Cruz, Alfredo Trejo-Martínez, Daniel Martínez-Carbajal, Tanya Arenas-Resendiz

Research output: Contribution to journalArticlepeer-review


This paper presents a general analysis of all the quartic equations with real coefficients and multiple roots; this analysis revealed some unknown formulae to solve each kind of these equations and some precisions about the relation between these ones and the Resolvent Cubic; for example, it is well-known that any quartic equation has multiple roots whenever its Resolvent Cubic also has multiple roots; however, this analysis reveals that any non-biquadratic quartic equation and its Resolvent Cubic always have the same number of multiple roots; additionally, the four roots of any quartic equation with multiple roots are real whenever some specific forms of its Resolvent Cubic have three non-negative real roots. This analysis also proves that any method to solve third-degree equations is unnecessary to solve quartic equations with multiple roots, despite the existence of the Resolvent Cubic; finally, here is developed a generalized variation of the Ferrari Method and the Descartes Method, which help to avoid complex arithmetic operations during the resolution of any quartic equation with real coefficients, even though this equation has non-real roots; and a new, more simplified form of the discriminant of the quartic equations is also featured here.

Original languageEnglish
Article number2377
Issue number14
StatePublished - 1 Jul 2022


  • algebraic equations
  • biquadratic equations
  • multiple roots
  • quartic equations
  • Resolvent Cubic


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