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

In order to propose a deeper analysis of the general quartic equation with real coefficients, analytical solutions for all cubic and equations were reviewed here; then, it was found that there can only be one form resolvent satisfies following two conditions at same time: (1) Its discriminant is identical equation. (2) It has least positive root whenever non-biquadratic. This unique special defined here as “Standard Form Resolvent Cubic”, which becomes relevant since allows us reveal relationship between nature roots forms cubic. Finally, this new basis designing programming efficient algorithms analytically solve algebraic fourth lower degree always avoiding application complex arithmetic operations, even when these have non-real roots.