Search results
Results from the WOW.Com Content Network
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 ...
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.
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.
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.
Its true value is the real solution of the equation x 3 = 2x 2 ... for the reciprocal root is the depressed cubic + ... solution with Cardano's formula ...
which is a depressed quartic equation. If ... (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 ...