In his groundbreaking text Ars Magna (Nuremberg, 1545), the Italian mathematician

Chapter 7, Problem 50

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In his groundbreaking text Ars Magna (Nuremberg, 1545), the Italian mathematician Gerolamo Cardano explains how to solve cubic equations. In Chapter XI, he considers the following example:X3 + 6x = 20.a. Explain why this equation has exactly one (real) solution. Here, this solution is easy to find by inspection. The point of the exercise is to show a systematic way to find it. b. Cardano explains his method as follows (we are using modem notation for the variables): 1 take two cubes v3 and u3 whose difference shall be 20, so that the product vu shall be 2, that is, a third of the coefficient of the unknown x. Then, I say that v u is the value of the unknown jc . Show that if v and u are chosen as stated by Cardano, then x = v u is indeed the solution of the equation jc 3 + 6x = 20. c. Solve the system1 - u3 = 20 vu = 2to find u and i>. d. Consider the equationx3 + px = qywhere p is positive. Using your work in parts (a), (b), and (c) as a guide, show that the unique solution of this equation is( D M ! ) '-V-W(!)+( ?) This solution can also be written as(!)2+(!)3What can go wrong when p is negative? e. Consider an arbitrary cubic equationjc 3 + ax2 + bx + c = 0.Show that the substitution x = t (a/3) allows you to write this equation ast3 + pt= q.