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3. Let the equation x3 -6x2=10, be transformad iato another, that shall want the second term.

Ans. y3 – 12y=26 4. Let y3 – 15y2 +81y=243, be transformed into an equation that shall want the second term.

Ans. 23+6x=88 3

7 9 5. Let the equation 23+ 2x2 + =0, be trans

8 16 formed into another, that shall want the second term.

11 3 Ans. y3 tay

16 4 6. Let the equation 2x2 – 3x2 +42—5—0, be transformed into another, that shall want its second term.

OF THE SOLUTION OF CUBIC EQUATIONS.

RULE

Take away the second term of the equation when necessary, as directed in the preceding rule. Then, if the numeral coefficients of the given equation, or of that arising from the reduction above mentioned, be substituted for a and b in either of the following formulæ, the result will give one of the roots, as required.

3+ ax=b
62

b 62
+
+ +

+
27

+

or

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62 a3

들이
+ ve +
4
27

b ba a3

tvat 2 4

27 Where it is to be observed, that when the coefficient a, of the second term of the above equation, is negative, a3 27" 3

as also , iņ the formula, will be negative ; and if

the absolute b be negative, », in the formula, will, also, be

b .

62 negative; but will be positive. (e)

4 It may, likewise, be remarked, that when the equation is of the form

x3 - Ax -+b

(e) This method of solving cubic equations is usually ascribed to Cardan, a celebrated Italian analyst of the 16th century ; but the authors of it were Seipio Ferreus, and Nicolas Tartalea, who discovered it about the same time, independently of each other, as is proved by Montucla, in his Historire des Mathematiques, Vol. 1. p. 568, and more at large in Hutton's Mathematical Dictionary, Art. Algebra.

The rule above given, which is similar to that of Cardan, may be demonstrated as follows: Let the equation, whose root is required, be x3 + ax=b.

And assume y +2=x, and 3yz Then, by substituting these values in the given equation, we shall have y3 +3y2z+-3y22 +23+uX(y+z)=y3 +z3+3y2x(y+8)+ ax (y+2)=y3+23-ax(4+2)+ax(+z)=b, or

y3 +235) And if, from the square of this last equation, there be taken 4 times the cube of the equation yz=-ga, we shall have y6-2y323 +36-b2 +2523, or

y3–23=v(ba+ya3) But the sum of this equation and y3+z3=b, is 2y3 =b+v(62+ a3) and their difference is 2x3=6-v(62+43); whence y= 36+ (+62 +27a3),

and z=36v(482 tazas). From which it appears, that ytz, or its equal xs is = 16+ (762+1793)+46-(162 + 7a3), which is the theo

rem.

a

Or, since z is= it will be y+z=y-, or x=
Зу

3y 3 16+(102 + 43

the same as the

3 36+ (163 +43 rule.

ģa

Q3

62 and

is greater than or 423 greater than 27b2, the 27

4 solution of it cannot be obtained by the above rule ; as the question, in this instance. falls under what is usually called the Irreducible Case of cubic equations. (f)

EXAMPLES.

1. Given 2x3 — 12x2 +36x=44, to find the value of ..

Here x36x2 + 18x=22, by dividing by 2.
And, in order to exterminate the second term,

6
Put x=zt

=2+2,

3
(2+2)3=23 +62 +12z+8
Then -6(2+2) 622 – 242-24

=22
18(2+2) =

182+36

X=

Whence 28 +6+20=22, or 23 +62=2, And, consequently, hy substituting 6 for a, and 2 for 6, in the first formula, we shall have, 2 4 216

2 4 216

); 4 27

4 27
Ýit(1+8)+/-v(1+8)=3/1+9+

V1-V1+3+31-3=>4-32,
Therefore x=2

z+2=/4-1/2+2=2+1.587401 1.259921=2.32748, the answer.

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(f) It may here be farther observed, as a remarkable circum. stance in the history of this science, that the solution of the Irreducible Case above mentioned, except by means of a table of sines, or by infinite series, has hitherto baffled the united efforts of the most celebrated mathematicians in Europe ; although it is well known that all the three roots of the equation are, in this case, l'eal ; whereas, in those that are resolvable by the above formula, only one of the roots is real, so that in fact, the rule is only applia cable to such cubics as have two impossible roots.

2. Given #3 - 6x=12, to find the value of x.

Here a being equal to -6, and b equal to 12, we shall have, by the formula,

-2 x=96+ (36-8)

v{6+7(34 —8)}

2
16+28+

=»(6+5.2915)+
V(6+728
2

2
* 11.2915+

= 2.2435+ (6+5.2915)

(11.2915) 2

=2.2435+.8957=-3.1392 2.2435

Therefore x=3.1392, the answer. 3. Given x3 – 2x=-4, to find the value of x.

Here a being -2, and b=-4, we shall have, .by the formula, 8

8

or 27 10

10

== {-2+v(4-1)}+V4-2-v(4-) },

3-2+1.9245 – /2+1.9245 =

=-.0755
3/3.9245=-.4226_1 5773=- 1.9999, or - 2

Therefore x=- -2, the answer. (8) Note. When one of the roots of a cubic equation has been found, by the common formula as above, or in any other way, the other two roots may be determined, as follows :

Let the known root be denoted by r, and put all the

(8) When the root of the given equation is a whole number, this method only determines it by an approximation of 9s. in the decima part, which sufficiently indicates the entire integer; but in most instances of this kind, its value may be more readily found, by a few trials, from the equation itself.

terms of the equation, when brought to the left hand side, =0; then if the equation, so formed, be divided by & Fr, according as r is positive or negative, there will arise a quadratic equation, the roots of which will be the other two roots of the given cubic equation.

4. Given x3 – 15x=4, to find the three roots, or values of x.

Here x is readily found, by a few trials, to be equal to 4, and therefore

3-4)33 -- 15x-4(22+4x+1

x34x2

4.x2 -15.2 4x2 - 160

I-4

- 4

*

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Whence, according to the note above giren,

22+41+1=0, or x2 +4x=-1; the two roots of which quadratic are - 2+13 and 2 -3; and consequently

4, -2+73, and -2-V3, are the three roots of the proposed equation.

5. Given x3 +3x2 - 6x=8, to find the root of the equation, of the value of x.

Ans. r=2 6. Given x3+x=500, to find the root of the equation, or the value of x.

Ans. =7.617 7. Given x3-48x2=

200, to find the root of the equation, or the value of x.

Ans X=47 9128 8. Given x3.- 6x=6, to find the root of the equation, or the value of x.

Ans. I=74+V2 9. Given x3 +-9x=6, to find the root of the equation, or the value of x.

Ans. I=/9-13

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