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The formula applies to depressed cubics, but, as shown in § Depressed cubic, it allows solving all cubic equations. Cardano's result is that if + + = is a cubic equation such that p and q are real numbers such that + is positive (this implies that the discriminant of the equation is negative) then the equation has the real root +, where and ...
In all, Cardano was driven to the study of thirteen different types of cubic equations (chapters XI–XXIII). In Ars Magna the concept of multiple root appears for the first time (chapter I). The first example that Cardano provides of a polynomial equation with multiple roots is x 3 = 12 x + 16, of which −2 is a double root.
Cardano's formula for solution in radicals of a cubic equation was discovered at this time. It applies in the casus irreducibilis , but, in this case, requires the computation of the square root of a negative number, which involves knowledge of complex numbers , unknown at the time.
Gerolamo Cardano was the first European mathematician to make systematic use of negative numbers. [12] He published with attribution the solution of Scipione del Ferro to the cubic equation and the solution of Cardano's student Lodovico Ferrari to the quartic equation in his 1545 book Ars Magna, an influential work on
This is a cubic equation in y. Solve for y using any method for solving such equations (e.g. conversion to a reduced cubic and application of Cardano's formula). Any of the three possible roots will do.
There are conjectures about whether del Ferro worked on a solution to the cubic equation as a result of Luca Pacioli's short tenure at the University of Bologna in 1501–1502. Pacioli had previously declared in Summa de arithmetica that he believed a solution to the equation to be impossible, fueling wide interest in the mathematical community.
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.
Widespread stories that Tartaglia devoted the rest of his life to ruining Cardano, however, appear to be completely fabricated. [24] Mathematical historians now credit both Cardano and Tartaglia with the formula to solve cubic equations, referring to it as the "Cardano–Tartaglia formula".