=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

Question
Asked Nov 5, 2019

#3 a and b

=
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
++
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= 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 ++

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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
help_outline

Image Transcriptionclose

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

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check_circleExpert Solution
Step 1

A] 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
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A] 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

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Step 2

Then
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 .
=
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Then 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 . =

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Step 3

...
Write 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)
help_outline

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Write 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)

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