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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.
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 ...
Its true value is the real solution of the equation x 3 = 2x 2 + 1. The name supersilver ratio results from analogy with the silver ratio , the positive solution of the equation x 2 = 2 x + 1 , and the supergolden ratio .
Scipione del Ferro was born in Bologna, in northern Italy, to Floriano and Filippa Ferro.His father, Floriano, worked in the paper industry, which owed its existence to the invention of the press in the 1450s and which probably allowed Scipione to access various works during the early stages of his life.
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 ...
The Cardano formula of this depressed cubic equation appears to be able to handle most every combination of the equation coefficients p and q. For example, with the depressed equation if the term (q^2/4+p^3/27) is negative, the use of complex numbers is most interesting as it offers the complete solution only requiring sorting through the ...