How to solve third degree equations without moving to complex numbers

Abstract

During the Renaissance, the theory of algebraic equations developed in Europe. It is about finding a solution to the equation of the form
a_nxⁿ + . . . + a₁x + a₀ = 0,
represented by coefficients subject to algebraic operations and roots of any degree. In the 16th century, algorithms for the third and fourth-degree equations appeared. Only in the nineteenth century, a similar algorithm for the
higher degree was proved impossible. In (Cardano, 1545) described an algorithm for solving third-degree equations. In the current version of this algorithm, one has to take roots of complex numbers that even Cardano did
not know.
This work proposes an algorithm for solving third-degree algebraic equations using only algebraic operations on real numbers and elementary functions taught at High School.