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

Abstract

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.

On the Practicality of the Analytical Solutions for all Third- and Fourth-Degree Algebraic Equations with Real Coefficients

“…Likewise, according to [6], the forms of the resolvent cubic with a 0 = 0 in Equation (18) are considered as "the translated forms of the Resolven Cubic", whereas the forms with a 0 = 0 are considered as "the non-translated forms of the Resolvent Cubic"; at first sight, all these considerations might seem superfluous, but they become relevant since the SFRC always has at least one positive real root for any Ferrari Case, as also stated in [6]; so, according to Equation (19), this fact also holds for the non-translated forms of the resolvent cubic whenever a 1 > 0; however, Equation (19) also implies that this fact does not necessarily hold for the translated forms of this equation.…”

“…r = 16ab 2 c − 64a 2 bd − 3b 4 + 256a 3 e 256a 4 ; (6) therefore, as stated in [6], Equation (5) allows us to classify the GQE in the following two complementary cases:…”

“…Since Equations (7) and the Ferrari Method are not complete general solutions of the GQE, a generalized variation of this method was developed in [6], which is a complete general analytical solution for both cases of the GQE; so, this general solution of the GQE depends on the roots of the SFRC, which is the third-degree equation…”

“…where s = 0 whenever p = q = r = 0 (in other words, when S = ∅); otherwise, s ∈ S = ∅; and ξ := ±1. Hence, the definitions of S and α s guarantee that Equation ( 9) never goes undefined, unlike the original Ferrari Method, which becomes undefined for the biquadratic case; on the other hand, as also stated in [6], the respective discriminants of the GQE, the DQE and the SFRC are all identical to each other, so the most simplified form of the discriminant of these three equations is given in terms of the coefficients of the DQE as follows:…”

“…Finally, Section 3 focuses mainly on the development of a computing program that was made using the Wolfram Mathematica programming language to analytically solve and plot all the polynomial equations of the second, third and fourth degree with real coefficients, based on algorithms designed according to all the results exposed in Section 2 and also by applying the formulae that instantly solve the non-biquadratic quartic equations with real coefficients and multiple roots that were also stated in [6].…”