PART XII. Of feveral Methods of Solving High Adfected Equations. CHAP. I. Some Preparations fometimes convenient for the Solution of High ADFECTED EQUATION S. TO Increase or Diminish the Value of the unknown Roots of an Equation by any given Quantity. RULE. Inftead of the unknown Root, fubftitute another unknown Root, Lefs or Greater than it, by the Quantity given. Thus, Example I. To Augment the Roots of this Æquation a a baddo, by the Quantity c. Put aec, then a abaddo is = e2-2 ec+cc+be-bcddo. Whofe Roots (e) exceed them of the former, (a) by the Quantity ; that is, eca. C Example II. To diminish the Roots of this Equation a2+ baddo, by the Quantity c. Putea, and then you'll have this Equation. ee + zec + cc + be + bc ddo. Whofe Roots are lefs than thofe of the Equation propos'd, by the Quantity c. SCHOLIUM I. By this manner of Subftitution, any Term, except the first of any given Æquation, may be destroyed. So if the Equation given be x4 - b x 3 + c x2 - dx+ f=0. Put ynx. Then 3 34 + 4ny3 + 6 n2 y2 + 4 n3 y+n+= xxxx cy 2 + z cny + cn2 = + cxx Now 'tis plain, that any Term, except the first, may be taken away from this Equation, because n was taken at pleafure. Vix. Putting 4 n b = o, the fecond Term muft vanish; and putting 6 n23bno, the third Term will vanish. In like manner any other Term in this, or any other Equation may be deftroyed, except the first. Therefore in taking away the fecond Term of any Equation, let the Index of the higheft Power of the fought Root x bem; and fuppofing the Coefficient of the second Term in the given Æquation, and you'll have another, in which y will be the Root fought, and which will want the fecond Term. Then after you have found the Value of y, you may find that of x, for y=x± a m SCHOLIUM II. By this way of Subftitution you may add to any Æquati on any Term that it wants. PREPA As fuppofing the Equation to be x4+ 3 x — d4 = 0. To Multiply or Divide the Roots of an Æquation by any given Quantity: Suppofe q. RULE. Inftead of the unknown Root fubftitute part, or q times, another unknown Root: Thus, I Example I. To multiply the Roots of this Æquation x3 + C 2 x = g3 by q. bx 2. And Multiplying each Part by q3 we have 23 + qb z z +92 c2 2q 3 g 3 an Equation wherein z isqx in the former.. > The Reafon is obvious, for if bex, then z = qx. X 3 1 Example II. To Divide by 3 the Roots of this Equation, 54c2 x — 216 d3 = 0. Suppose 3 x = x, then the former Aquation will become 27 23 162 0 22 -- ·216 d3 = 0. a And by Dividing each Part by 27, we have z3-6c2z x 8d3o; an Equation wherein z is = in the for 3 mer COROLLARY. Hence an Equation may be clear'd from Fractions, without increafing the Coefficient of the higheft Term; thus, if 40 7 to a common Denominator) if x 3. + 9261 441 =0; and by Mul 21 tiplying each Part by 9261, we have 23+147252920=Q. See the Ufe of this Corollary in Page CHA P. II. The Solution of CUBICK EQUATIONS, by Cardan's Method. WHen any Cubick Equation having the fecond Term, be propos'd to be refolv'd, by Cardan's Method, you must first deftroy its fecond Term (by the laft Chap. Propofition I. Scholium I.) and then you'll reduce it into one of these four following Forms or Cases, Viz. In either of thofe Equations the Quantity fought (a) is Inferted only in two different Terms, in which its Indices are triple to one another: And in order to folve all Equations thus thus qualified at once, let us fuppofe the reduc'd Equation to be equal to x2 + bx 3 = c, in which the Value of b and e may be either equal to any given Quantity whatsoever, Affirmative or Negative; and that of n is indetermin'd, but is in all original Cubick Equations = 3; and generally in all other Cubicks, equal to fome Multiple of 3, and x" is understood to have the Sign + prefix'd to it. Then 5 e3 + zeey + zeyy+y3+bxe+y. (= 23+ bx) = c. 3 Now fuppofe6b3ey From 5 and 6 5 6÷ 6 ÷ 3e 8 3eyy=ces+ys 803 9 2763 b3 |