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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.
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".
The cubic-plus-chain (CPC) [28] [29] [30] equation of state hybridizes the classical cubic equation of state with the SAFT chain term. [21] [22] The addition of the chain term allows the model to be capable of capturing the physics of both short-chain and long-chain non-associating components ranging from alkanes to polymers. The CPC monomer ...
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.
Lodovico settled in Bologna, and he began his career as the servant of Gerolamo Cardano. He was extremely bright, so Cardano started teaching him mathematics. Ferrari aided Cardano on his solutions for biquadratic equations and cubic equations, and was mainly responsible for the solution of biquadratic equations that Cardano published. While ...
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.