Thus, the term "theory of equations" is mainly used in the context of the history of mathematics, to avoid confusion between old and new meanings of "algebra".
Nevertheless, the main concern of the algebraists was to solve in terms of radicals, that is to express the solutions by a formula which is built with the four operations of arithmetics and with nth roots.
Scipione del Ferro and Niccolò Fontana Tartaglia discovered solutions for cubic equations.
Gerolamo Cardano published them in his 1545 book Ars Magna, together with a solution for the quartic equations, discovered by his student Lodovico Ferrari.
In 1572 Rafael Bombelli published his L'Algebra in which he showed how to deal with the imaginary quantities that could appear in Cardano's formula for solving cubic equations.