  # =xtraor-ld be engendered by imaginary numbers. From this timeof their mystical character, although their full acceptancethe roots of the cubic xin the 1800s.136r2= 20x + 56.(c)3(d) x64 6x2+ 24x.2. Derive Cardan's formulap'3+27q2V 43q3qV 42227for solving the cubic equation x3p>O and q> 0.= px +q, where123. Using Cardan's formula, obtain one root of each of the++ Cardan's Ars Magnafollowing cubic equations.HIB(a) x3+24x 16.(b) x315x6x218(c) x327(d) x39x 12.6x2+15x +83x2+27x+ 41.AN6x2+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, eachwhom divided 48 aurei among his soldiers. One ofthese had two more soldiers than the other. The onewho had two soldiers fewer had four aurei more foreach soldier. Find how many soldiers each had.6.Chapter 37, Problem 1. The dowry of Francis's wife100more than Francis's own property is worthquare of the douaureiand thethan the100noreSS

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#3 a and b help_outlineImage Transcriptionclose= 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 ++ fullscreen help_outlineImage TranscriptioncloseCardan'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 fullscreen
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Step 1  help_outlineImage TranscriptioncloseA] The cubic equation is x' +24x = 16 By Cardan's method: Consider x v+ w and take cube on both sides. x w3 (v+ w 3vwx + (v +w On comparing 3vw-24 vw-8 vw3=-512 w16 fullscreen
Step 2  help_outlineImage TranscriptioncloseThen v16, w 32 v = -22, w 2/4 = 23/4-2 /2 W Thus, x By Cardan's formula: On comparing, p -24 and q 16 Thus, the value ofx becomes 16) (-24) 16) (16) (-24) (16) 2 4 27 2 27 + 64+512 8- 64+512 85768-576 V8+248-24 = That is x 2/4-2/2 Therefore, one ofthe roots for cubic equation is 2{/4 - 2{/2 . = fullscreen
Step 3 ... help_outlineImage TranscriptioncloseWrite the equation as x'-6x2 +15x-18=0 www. Remove the x term as follows: а, wheren is the highest degree ofthe equation which is 3 here, a, is па Consider h= the coefficient of x' which is 1 here and a, is the coefficient of x which is -6 here (-6) = 2 Thus, the value ofh is h (3)(1) fullscreen

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