A New Method of Resolving Cubic Equations

A New Method of Resolving Cubic EquationsAuthor(s): Thomas MeredithSource: The Transactions of the Royal Irish Academy, Vol. 7 (1800), pp. 69-77Published by: Royal Irish AcademyStable URL: http://www.jstor.org/stable/30078942 .

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69

A new METHOD of refolving CUBIC EQUATIONS,

By THOMAS MEREDITH, A. B. Trinity College, Dublin.

THE roots of a cubic equation of this form, x3 3 c. xt 3 0. x Read June 6.3-a 0 which differs from a power only in its laic term loth, 1797

can be found, by tranfpofing, a, and extracting the cubic root ou

each fide, provided, a, is not an impoffible binomial.

PROBLEM, To reduce any cubic equation to this form, x3 3 C. X 3 C. x c' -a a, that is, to reduce it to an equa tion, in which, the fquare of the co-efficient of the fecond term is

triple the co-efficient of the third.

IF the roots of a cubic equation, x3 +px qx r o, are en

creafed or diminifhed by any quantity, p2 and 3 q, will be en

creafed or diminiihed by an equal quantity, if multiplied, will

be multiplied by an equal quantity, therefore their equality or

inequality, not affecied by thofe transformations.

X



70 I

j3 3 ay2 3 ey -1-a3 2p?ai'+paz

f.Y-Fga r

Y +a

r

p' =9 iv 6 ap +p and 3q. gat 6 ap 3 7 thereforep and 3 q. encreafed by the fame quantity, viz. 9 a 6 ap

x3 +pie +9,x+r=a

14 +az qa' r o a

pZ a2 and 3 g 3 q az therefore both multiplied by the fame

quantity, viz. a'.

HENCE it appears that the problem cannot be efFeded by thofe transformations.

BUT the equation, ex3 -Fpx' --Fqx-Fr=a by transforming the roots into their reciprocals, and freuing the firft term from a coeffi cient becomes, x' qx2 +prx rz therefore if in the propofed equation 3pr, then by transforming the roots into their re

ciprocals, and freeing the firft term from a co-efficient the equatioa will be reduced to the required form.

ANY cubic equation being propofed, there is a quantity, by which if the roots are encreafed (or diminiiheci 9'2 will become

equal



71 I

equal 3p 7., the value of this quantity may be inveiligated by

folving a quadratic equation.

THUS let the equation be, x3 -Fpx2 qx+r=o

j3 yyz 3 ezy t e3

x=y-f-e Ey' -I 2 Pe, pez

Ty-1-v -Fr

___;2,-..-..,..,, +6T 3e -1-2pc+q-1- 9e4i12pe 3 . e2r 41'9re qi

+47

3+97 +9r 3x0+Ptz-Eqe+rx-Te9e4+12pe .ez . e 3pe

31'2 3154

+6g +97 +9r 9e4.-1-12pe3 . e' vie 93 9e4+12,pe3 .ez . e 3pr

+4P +3P2 +3.Pg

P2 +1,9" +9'31 .e2 .e =-_,(3

-3q -9r -3pr

LET it be required to find the roots of this equation, x3 6x2 3 x 2 0. Subilituting in the formula

36 +18 9 .e 2 . e =o... 27 e 27

. .. ez I *. e I. 9 -18 36

oel



72

X3 6,10 3X 2=0 X3 Y3 3.Y1 3.Y 1

,t =y i 6y 12y 6

3.Y 3 2

y3 +9Y1 183 12 =0

/27)3 18 VI 9V 1 --=-0

z3 /8 z. 4. IC8 Z 44 0 Extract the cubic

root on each fide 1 Z3 18 zs '08 z 216 72

Z 6 =4/72 V.Z.--., 6 2%/9

1 V 7 6 2 3/9 -. 2 12

3. 1 .... 12 11-1-"' 6 2 4/9

6 2 9

fubftituting then for 2 43/9 its 3 values, 2 Z/9, -I 4/-3 x v9,

-I.-v.-3 x 9 the roots of the propofed equation will be

12 12 12 . -1 Il f----- 3 15 e" 3 I

-6 2 ;f9 -6 "-I V-3. v9 .-6-1----v-3. V9

BUT the roots of the given equation may be found after one

transformation, for the roots of the final equation are expreffed

in the co-efficient of the firft transformed equation, and the

root



73 :I

toot of the firft transformed is its abfolute quantity divided 1). the root of the final equation.

LET the equation when its roots are encreafed by e, he I

)0 +19,3 -1-gy r ==. o )1 = 7i; z

1)= r r v3 g. v2 -Fp v r o -3 9 z' r pz r2 0

3 ....-. q 3 ., ---r

Z ..... I sh' 3 _ri v =_.

v 27 T 3 V 27

r

y /9,----i -- .r-b.......

liv 7 a 2 ,

WHEN the value of e, by which the roots are to be encreafed

(or diminifhed is impoffible the coefficients of the transformed

equation will be impoffible binomials h. a 13 --r' an impoffible ..27

binome (unlefs in particular cafes the coeff. of the impoffible part

vanishes hence it appears that a, and e, will be poffible or im

poflible in the fame cafes.

Vol,. VII. K Ir

: Since the equation z3 q zt -I- rpz rt may be thus exprefred z3 +3. 5.-z2 3. f. z 1.3. V .. r'.

3 9 27 27



74 1

IF the roots of a cubic equation are encreafed by 3 the fecond

term will vaniih, if by 3 v 9 3 --9 the third term will vaniih,

therefore ifp 3 y the fecond and third terms may be exterminated

together, therefore the equation will have two impoffible roots ;

hence it appears that the equation of the required form has two

impoffible roots, confequently the value of e, by which the roots are

to be encreafed will be impoffible when all the roots of the propofed

equation are real a, whofe cubic root muff be computed, will be

an impoffible binomial,, unlefs in the particular cafes where the

coefficient of the impoffible part vanifhes.

IT remains to be proved that when the propofed equation has

but one poffible root, the value of e, by which the roots are to be

encreafed (or diminifhed will be poilible and confequently a,

LET the roots be m n, -m v n, -6

p 2 M 62 m. n B fit, r brna bn

, -FP9' +9"1 . .e

-3q -9r -3pr m2.

That the equation of,the required form has two impoffible roots, appears alto from this, that two of the cubic roots of a, are impoffible.



r 75

m.. e2 2 m3 e m4

2 bm 4bm2 zb

3n 2 mn b2

62 2 62 m dinn 8 bn -1 2 nrnz

bl n

Call the coefficients j3, 5,, e 212 4441 IG 2

therefore it is to be proved that 4

(3 3 is affirmative, and confe

quently the fquare root poilible.

4 711 -4bms 4-2n. m4 -12bn. m3 +1'8P n. m2 -8bnz.m-1-1662n2

+662 4b3 122 1.4

m4-26m3+6 m2-26mn+znmi-362xte-zbm-3n+bz

ins -4bmg +662.n:4-463.W -5nz. m2 41722. m-f rob2 n2 b4. +41.3 n 3n3

3b4n

pa.312. -i2bn.m3+18Pn.m2-nbn2.7/2+6Pn2

6 nz -I2b'n 3n3 364 tz

K 2 but

2, 5' =m3 -abm2 +mnd-b2m 4671



76

but this remainder may be refolved into thofe parts.

m4 4:3-M3 6 b2m2 4 b3 #2-Wx 3 n m--6.4 3 n

#12-24m+b2x6e m=b21. 6 n3

3n3 3n3

n, is affirmative, and m-44 and m---62 alto affirmative therefore

the remainder affirmative.

LET the roots be m

any'. e2 m2. 2 M. c .

-2bm -2bm 2 b m =o

62. +b2

e2 2m. e o

e m 7-1-7 m2

therefore when two roots of the propofed equation are equal, the value e, by which the roots are to be encreafed will be one of the

equal roots, therefore the two daft terms of the transformed equation will vanifh, therefore reducible to a Pimple equation, which will give the remaining root e.g. x3 7 OC2 16 x I 2 o

49 112 256 e2 e =o...e2-1-4e+4=ove=-2qV 4-4

-48 108 252 3 X



77

x3 .y3 6y2 I 2, 8

N=Y+2 7,x2 7)12. 28y 28

16 x 6y 32

12

Y3 +.)/ =0 2



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A New Method of Resolving Cubic Equations