6 Since:we get:Since y = z + (p/3z), we get the solution:
7 Fromandthe real (since p, q are real) solution to the reduced equationy3 + py + q = 0is:
8 Relabel:So if y3 + py + q = 0, we can rewrite the real solution as:But there should be two others!
9 By DeMoivre’s Theorem, if A is the real cube root of: then are the other two (complex) roots, where:Similarly, since B is the real cube root ofthen the other two (complex) roots are:
10 .The resulty = A + Bas a solution of:y3 + py + q = 0will also hold if A and B are replaced by the other respective cube roots, so long as the product of the terms is AB = p/3.Since 3 = 1, this will also hold true for the pairs
11 We get that the three solutions of: y3 + py + q = 0are:where:Add b/3 to each solution to get the solutions of:x3 +bx2 + cx + d = 0.
12 Example: x3 3x + 2 = 0The equation is in reduced form with p = 3 and q = 2.So x1 = 2 is our first solution.
13 Using our previous notation, x1 = A + B, where A = B = 1. The second solution is x2 = A + 2B, whereThus:Similarly the third solution x3 = 2A + B = 1.Thus the three solutions of x3 3x + 2 = 0 are: x = 2, 1, 1.
14 Example x3 +12x2 + 54x + 68 = 0Here we have b = 12, c = 54, and c = 68.We transform to reduced form by letting x = y b/3 = y 4.Reduced Form: y3 + 6y 20 = 0. (Cardano’s Example)Here p = 6 and q = 20.
19 Thus the three solutions of y3 + 6y 20 = 0are:y = 2, 1 3i.Remember that x = y 4. Thus the solutions to the original equation:x3 +12x2 + 54x + 68 = 0
20 Chronology of the Cubic In his book Summa de Arithmetica, the Italian LucaPacioli states that, given the state of mathematics, thegeneral cubic ax3 + bx2 + cx + d = 0 is unsolvable.c Scipione del Ferro solves x3 + px = q, x3 + q = px andx3 = px + q, where p, q > 0. He does not publish it.c Del Ferro reveal his result to his colleague AntonioMaria Fior.Fior challenges the Venetian mathematician NiccolòFontana (aka “Tartaglia”) to a contest where each posesthirty math problems to the other. The loser is to payfor a dinner for thirty. All of Fior’s problems involvethe above cubics.
21 Chronology of the Cubic On the night of February 1213, Tartaglia solves allthirty problems. He generously passes on the dinner.The scholar Girolamo Cardano asks Tartaglia (through athird party) what his solution to the cubic is. Tartagliarefuses. Cardano then invites Tartaglia to Milan as hisguest. He offers Tartaglia the opportunity to show off hismilitary inventions to the commander of Milan. (He alsogives Tartaglia a dinner.)On March 29, Tartaglia reveals his solution to x3 + px = qto Cardano. He swears Cardano to an oath of secrecy.Cardano extends the result to general cubic equations.
22 Chronology of the Cubic Cardano publishes Tartaglia ’s result, along with that of hisprotégé, Ludovico Ferrari. Ferrari shows how to use thereduced cubic to solve quartic equations. Cardano citesdel Ferro and Tartaglia in his treatise.A furious Tartaglia sues. Ferrari issues a public challenge.The case is argued in a Milan court on August 10th.Tartaglia leaves before the case is settled. He loses.A group of scholars at a KY teaching conference learnsabout Cardano’s (and Tartaglia’s, and del Ferro’s) formula.
23 BibilographyDobbs, David and Hanks, Robert. A Modern Course on the Theory of Equations. Polygonal Publishing House,Dunham, William. Journey Through Genius: The Great Theorems of Mathematics. Wiley,Irving, Ron. Beyond the Quadratic Formula. MAA Inc., 2013