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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 ...
Furthermore, his student Lodovico Ferrari found a way of solving quartic equations, but Ferrari's method depended upon Tartaglia's, since it involved the use of an auxiliary cubic equation. Then Cardano became aware of the fact that Scipione del Ferro had discovered Tartaglia's formula before Tartaglia himself, a discovery that prompted him to ...
Casus irreducibilis (from Latin 'the irreducible case') is the name given by mathematicians of the 16th century to cubic equations that cannot be solved in terms of real radicals, that is to those equations such that the computation of the solutions cannot be reduced the the computation of square and cube roots.
However, in 1925, manuscripts were discovered by Bortolotti which contained del Ferro's method and made Bortolotti suspect that del Ferro had solved both cases. Cardano, in his book Ars Magna (published in 1545) states that it was del Ferro who was the first to solve the cubic equation and that the solution he gives is del Ferro's method.
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.
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
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 ...
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".