Where it is to be observed, that if the three roots of the cubic equation, r', r", r'", be all real and positive, the four roots of the proposed equation will, also, be real; and if one of these roots be positive, and the other two imaginary, or both of them negative, and equal to each other, two of the roots of the given equation will be real, and two imaginary; which are the only cases that produce real results. 3. Given - 25x2 + 60x−36=0, to find the four roots of the equation. Here a 25, b=60, and e=-36; Whence, by substituting these values for their equals, in the cubic equation above given, we shall have 2 × 25x2 + (252 + 4 × 36)≈≈ 60o, or 2-50%2+769x=3600: The three roots of which last equation, as found by trial, or by either of the former rules, are 9, 16, and 25, respectively; whence -6 x=(√9-√16 — √25) = (-3-4-5) x=(−√9+ √16 +√25) = ( − 3+ 4+ 5) = +3 x=1(+√9−√16 + √25) = ( + 3-4+5)= +2 x=(+√9+√16-√25)=(+3+4-5)= +1 And consequently the four roots of the proposed equation are 1, 2, 3, and -6. 1 EXAMPLES FOR PRACTICE, 1. Given x* - 55x2 - 30x + 504-0, to find the four roots, or values of x. Ans. 3, 7, 4, and -6 2. Given x+2x3-7x2-8x=-12, to find the four roots, or values of x. Ans. 1, 2, 3, and -2 3. Given x1-8x3 + 14x2 +4x=8, to find the four roots, or values of x. 4. Given - 17x2 - 20x-6=0, to find the four roots, or values of x. 5. Given x-3x2-4x=3, to find the four roots, 6. Given x-19x+132x2-302x + 2000, to find the four roots of the equation. 7. Given x*-27x3 + 162x2 + 356x-1200=0, to find the four roots of the equation. 8. Given x-12x2 + 12x-3=0, to find the four roots of the equation. Ans. {{ .606018, -3.907378 2.858083, .443277 9. Given x-36x2 +72x−36=0, to find the four roots of the equation. 10 Given 1.2679494 0.8729836, Ans. 4.7320506, -6.8729836 -5x+13x2-17x+12=0, to find the four roots of the equation, which are all imaginary. OF THE RESOLUTION OF EQUATIONS BY APPROXIMATION, OR THE METHOD OF SUCCESSIVE SUBSTITUTIONS. (s) EQUATIONS of the fifth power, and those of higher dimensions, cannot be resolved by any rule or algebraical formula, that has yet been discovered; except in some particular cases, where certain relations subsist between the coefficients of their several terms, or when the roots are rational, and, for that reason, can be easily found by means of a few trials. In these cases, therefore, recourse must be had to the following method of approximation, which is universally applicable to all kinds of numeral equations, whatever may be the number of their dimensions, and, though not strictly accurate, will give the value of the root sought, to any required degree of exactness (c). (c) The first attempt to determine the roots of equations, by approximation, appears to have been made by Vieta, in his treatise de Numeros a potestatum adfectarum Resolutione, where he follows a process similar to that now used for obtaining the roots of numbers; which method was afterwards improved by Harriot, Oughtred, Pell, and others, who rendered it something more simple; but the multitude of operations that were required to be performed, and the uncertainty of success in a great number of cases, occasioned it to be entirely abandoned before the end of the seven RULE. Find, by trials, a number nearly equal to the root sought, which call r; and let ≈ be made to denote the difference between this assumed root, and the true root x. Then, instead of x, in the given equation, substitute its equal r+%, and there will arise a new equa-tion, involving only ≈ and known quantities. Reject all the terms of this equation in which ≈ is of two or more dimensions; and the approximate value of ≈ may then be determined by means of a simple equation. And if the value, thus found, be added to, or subtracted from that of r, according as r was assumed too little, or too great, it will give a near value of the root required. But as this approximation will seldom be sufficiently exact, the operation must be repeated, by substituting the number, thus found, for r, in the equation exhibiting the value of x; when a second correction of % will be obtained, which, being added teenth century. Since that time Simpson (in his Select Exercises) and several later writers, have employed different methods for the same purpose; but the rule given in the text, which is that of Newton, as improved by Ralphson, will be found as commodious, and easy in its application, as any that has yet been proposed. Its chief defect, which likewise attends the other methods here mentioned, is, that it does not show the progress made in the approximation at each operation; and that, when some of the roots are nearly equal to each other, they may be passed over without being perceived; both of which circumstances have been particularly noticed by Lagrange; who has given a new and improved method of approximation, in his late work de la Résolution des E'quations Numériques, to which we must refer the reader, to, or subtracted from r, will give a nearer value of the root than the former. And by again substituting this last number for r, in the above mentioned equation, and repeating the same process as often as may be thought necessary, a value of x may be found to any degree of accuracy required. Note. The decimal part of the root, as found both by this and the next rule, will, in general, about double itself at each operation; and therefore it would be useless, as well as troublesome, to use a much greater number of figures than these in the several substitutions for the values of r. EXAMPLES. 1. Given x2 + x2+x=90, to find the value of x, by approximation. Here the root, as found by a few trials, is nearly equal to 4. Let therefore 4=r, and r+z = x; Then x3 = r3 + 3r2 z + 3rz2 + ≈3 x2= r2+2rz + z2 x=r+% 2 1 =90. And by rejecting the terms 3 + 3r2 + z2, as smal! in comparison with %, we shall have 29 +22+r+3r2≈ + 2rz + %=90; And consequently x=4.1, nearly. Again, if 4.1 be substituted in the place of r, in the last equation, we shall have › as being but little susceptible of that kind of concise elucidation, which belongs to a performance like the present. |