Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

** + rx + o, we have x, in the one,

- 1

[merged small][merged small][merged small][merged small][ocr errors][ocr errors][merged small][merged small][merged small][merged small]

four roots of the biquadratic, x2 + ax2 + bx + €

[ocr errors][merged small][ocr errors][merged small][merged small][merged small][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small]

Let the equation propounded be y44y38y + 32 o; then, to take away the fecond term thereof, let xy; whence, by fubftitution, - 6x2·16x + 21 ±0; which being compared with the general equation, *** + ax2 + bx + c = 0, we here have a 16, and c = 21; and confe

6, b =

quently po — 12p* —48p2 (=p*+ 2ap+ ± aˆa } p2) =

4c

256 (62). Now, to deftroy the second term of this laft equation alfo, make z + 4 = p2; and then, this value being fubftituted, you will have z3- 96% 576; whence, by the method above explained, z will be found (288 + √ 288|2

[blocks in formation]

32/9/5

+ +

) 12. Therefore p (=

[ocr errors][merged small][merged small][merged small][ocr errors][ocr errors][subsumed][ocr errors][ocr errors][merged small][merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small]

— 9 = − 2 + √ — 3, and -- √PP

[ocr errors]

2

4

3; which are the four roots of the equation - 6x2-16x + 21; to each of which let ** unity be added, and you will have 4, 2, − 1 +

3,

and — 1 3, for the four roots of the equation propofed; whereof the two laft are impoffible. And that these roots are truly affigned, may be eafily proved by multiplying the equations, y 40, - 3 y − 2 = 0, y + I = √". 30, and y +I+V o, thus arifing, continually together; for, from thence, the very equation given will be produced.

The refolution of biquadratics by another method. In the method of Des Cartes, above explained, all biquadratic equations are supposed to be generated from the multiplication of two quadratic ones: but, according to the way which I am now going to lay down, every fuch equation is conceived to arife by taking the difference of two complete fquares.

Here, the general equation x + ax3 + bx2 + cx + d = o being propofed, we are to affume x2 + tax + A22- Bx + C2 = x2 + ax3 + bx2 + cx+d; in which A, B, and C, reprefent unknown quantities, to be determined.

Then, ax + A, and Bx + C being actually involved, we fhall have

x2 + ax3 + 2A+2

* + {a2x2+aAx + A2 ` = x2 + ax2 + bx2 2BCx C

*

--

+c+d: from whence, by equating the homologous terms, will be given, 1. 2A +

2. aA

3. A2.

a2. Bb, or, 2A + ža2 —b=

2BC

C&

= c, or, qA C

=d, or,

A2

d

B2;

2BC;

= C2.

Let now the first and laft of these equations be multi

plied together, and the product will, evidently, be

equal

equal to of the fquare of the second, that is 2A3 + aa—b x A2 — 2dA — d x aa—b (= B2C2) = 1 × a2A2—2acA + c2 (=B2C2). Whence, denot、 ing the given quantities ac-d, and 1 c2+dx±aa-b by k and 1, refpectively, there arifes this cubic equation, A3 — 1⁄2bA2 + kA — l = 0: by means whereof the value of A may be determined (as hath been already taught); from which, and the preceding equations, both B and C will be known, B being given from thence ➡ √2A+ aab, and C =

2

a A

- -C

2B

The several values of A, B, and C, being thus found, that of x will be readily obtained : for x2 +‡ ax + A}* -Bx+C being univerfally, in all circumstances of x, equal to x + ax3 + bx2 + cx+d, it is evident, that when the value of x is taken fuch, that the latter of thefe expreffions becomes equal to nothing, the former must likewife beo; and confequently

x2 + 1⁄2 ax + Al2 Bx+C: whence, by extracting the fquare root on both fides,x2+ 1⁄2 ax+A=±Bx±C; which, folved, gives x±Ba±√a{B}2±C—A

+ Ba±√za2 + AaB+1B2±C — A ; exhibiting all the four different roots of the given equation, according to the variation of the figns.

[ocr errors]

-

This method will be found to have fome advantages over that explained above. In the first place, there is no neceffity here of being at the trouble of exterminating the second term of the equation, in order to prepare it for a folution: fecondly, the equation A3 A + kA 11 = 0, here brought out, is of a more fimple kind than that derived by the former method: and, thirdly (which advantage is the most confiderable) the value of A, in this equation, will be commenfurate and rational (and therefore the easier to be discovered), not only when all the roots of the given equation are commenfurate, but when they are irrational and even impof fible; as will appear from the examples subjoined.

[blocks in formation]
[ocr errors]

Exam, 1. Let there be given the equation + 128 170.

Which being compared with the general equation x+ + ax3 + bx? + cx + do, we have a = 0, bo, c = 12, and d 17: therefore k (ac-d) = 17, ? (¦c2 + d × ¦aa — b) = 36; and confequently AA+A / A3 +17A 18 = 0; where = it is evident, by bare infpection, that A 1. Hence B (= √2 A + ¦aa —- b) = -- b) = √2, C(=

a A - C

) =

[blocks in formation]

2B

+3

= = = √ 2 ± √ √ = 3√ = = = = . Therefore the four

2

roots of the equation are ¦ √2 + √ √ — 3√2 — —

[ocr errors][merged small]

and - 2 - √ 3√ 2 — —; whereof the first and

fecond are impoffible.

x4.

Exam. 2. Let the equation given be x— 6x3

[blocks in formation]

58x*

[ocr errors]

Here a 6, b = 58,114; and d —— whence k (ad) = 182, 1(4cc + dx aa—b) = 2512; and therefore A3 +29 A+ 182A—1256 Q. Where, trying the divifors 1, 2, 4, 157, &c., of the laft term (according to the method delivered on p. 134) the third is found to fucceed; the value of A being, therefore, 4. Whence there is given B = ~75 = 5 31 go 33, and x * ( ± 1 B — ¦à ±

[ocr errors]
[ocr errors]

aB + B2± C — A) = ± √3 + 1 ÷

[ocr errors]

Exam. 3. Let there be now propofed the literal equation *+ + 2ax3-37a7%2 38a3% + a2 = 0.

This equation, by dividing the whole by a, and writing x = Z is reduced to the following numeral

a

[ocr errors]

one, x+2x3-37x2- 38x+1 o. If, therefore, a, b, c, and d, be now expounded by 2, 37,38, and I, respectively, we fhall here have k ( ac-d) = -20, 1 (c2 + d x aab)

by fubftituting these values,

399; and therefore,

[merged small][merged small][ocr errors][merged small][ocr errors][merged small]

Which equation, by the preceding methods, will be found to have three commenfurable roots,,3 and 19: and any one of these may be used, the refult, take which you will, coming out exactly the fame. Thus, by taking 3, for A, we fhall have x2 + x 3 = ± √2 × 4x+2: but, if A be taken, then will x2 + x + 1⁄2 = ± √5 × 3x+1: laftly, if A be taken —— 19, then x2 + x — ∙19 +6√10. All which are, in effect, but one and the fame equation, as will readily appear by fquaring both fides of each, and properly tranfpofing; whence the given equation x2 + 2x2 + 37x2 - 38x + 1 = 0, will, in every cafe, emerge. And the fame observation extends to all other cafes, where there are more roots than one; it being indifferent which value we ufe unlefs, that fome are to be preferred, as being the moft fimple and commodious.

Having given the general folution of_biquadratic equations, by the means of cubic ones, I fhall now point out two or three particular cafes, where every thing neay be performed by the refolution of a quadratic only.

Thefe

« ΠροηγούμενηΣυνέχεια »