= xtraor- ld be engendered by imaginary numbers. From this time of their mystical character, although their full acceptance the roots of the cubic x in the 1800s. 1 36r2= 20x + 56. (c) 3 (d) x64 6x2+ 24x. 2. Derive Cardan's formula p'3 + 27 q2 V 4 3q 3q V 4 2 2 27 for solving the cubic equation x3 p>O and q> 0. = px +q, where 12 3. Using Cardan's formula, obtain one root of each of the ++ Cardan's Ars Magna following cubic equations. HIB (a) x3+24x 16. (b) x315x6x218 (c) x327 (d) x39x 12. 6x2+15x +8 3x2+27x+ 41. AN 6x2+58. . AR (e) x3 = (f) x3= 4. Solve the cubic equation x+6x2+x =14. Problems 5-11 appear in Cardan's Ars Magna. 5. Chapter 5, Problem 2. There were two leaders, each whom divided 48 aurei among his soldiers. One of these had two more soldiers than the other. The one who had two soldiers fewer had four aurei more for each soldier. Find how many soldiers each had. 6.Chapter 37, Problem 1. The dowry of Francis's wife 100 more than Francis's own property is worth quare of the dou aurei and the than the 100 nore SS

Let AABC be a triangle. Fix the points X, Y, and Z on AB, AC, and BC, respectively (as in the figure) such that BX = - = CZ AY = 4. Assume that the area of AXYZ is one and calculate the area of AABC. ХА YC ZB

19 a-c

b. a = Vx-36, b = 6