Or the four roots of the given equation, in this last case, will be as follows: b b ) } + v{− 2 + 3 + v [(r+ {b}2 —d]} b {} + √ [(r + { b)3 — a] } b 3 3 Match 2. Given x4 + 12x - - 170, to find the four roots of the equation. Here a = 0, b = 0, c == 12, and d 17; Whence, by substituting these numbers in the cubic equation, 1 108 we shall have, after simplifying the results, 22+172 = 18, 1 1 b + c2 — } bd, 8 Where it is evident, by inspection, that z=1. And if this number be substituted for r, 0 for b, and 17 for d in the two quadratic equations in the above rule, their solution will give *C § x = −√2 + √ (− + √ 18) = −√2 + √(1+3√2) v √2 NI + √18) 1/ √ 2 ·√( + 3 √2) x = + + √2 + √(— — √18) = + + √2 + v( — — 3 √2) x = + + √2−√(— — √18) = + √2 −√ (− 1 − 3 √2) Which are the four roots of the proposed equation; the first two being real, and the last two imaginary. RULE* 3.-The roots of any biquadratic equation of the form x2 + ax2 + bx + c = 0, may also be determined by the 4 2 ratics, 22+px+9=0, and x2+rx+s=0, or its equal x2-px+s=0 we shall have. x = − } p±v († p2−q); x=\p ± √ (} p2 — §) ; which expression, when taken in and proposed biquadratic as was required. * This method, which differs considerably from either of the former, consists in supposing the root of the given equation, x++ax2+ bx +c=0(1), to be of the following trinomial surd form x = √ p + vq tv r ; where p, 4, 7, denote the roots of the cubic equation y3 + fy2+gy = h (2), of which the coefficients f, g, and the absolute term h, are the unknown quantities that are to be determined. give the four roots of the following general formulæ first given by EULER; which are remarkable for their elegance and simplicity. Find the three roots of the cubic equation z3 +2az2 + (a3 — 4c) z = = b2, by one of the former rules before given for this purpose; and let them be denoted by r', r', and '"' Then, agreeably to the theory of equations before given, we shall have p+9+r ·ƒ ; pq+pr+qr=g; pqr=h. And by squaring each side of the formula expressing the value of x, x2 = p + q + r + 2 v pq + 2 V pr +2 √ qr. Or, by substituting f for its equal (p+q+r), and bringing the term, so obtained, to the other side of the equation x2+ƒ= 2 √ pq+2 v pr+2 1/gr. Also, by again squaring each side of this last expression, we shall have x4+2fx2+f2=4pq+4pr+4qr+8 v p2 qr +8 √ q2 pr+8√ r2pq. Or, substituting 4g for its equal 4pq+4pr+4qr, and bringing the term to the other side as before, V x4 + 2ƒx2+ƒ2 → 4g = 8 √ pqr (Vp+vq+v r). But since, from what has been above laid down, we have vp + vq + v r = x, and pqr = √ h, if these be put for their equals in the last equation, it will become, by this substitution, x4 +2ƒx2 — 8h*x+ƒ2—4g = 0. Whence, comparing these coefficients with those of the given equation, there will arise 2f = a; a 8 √ h = b; f2—4g=c; or, v b2 a2 C h 2 16 4 ƒ= And consequently, by substituting these values in the assumed cubie equation (2), we shall have 64 b2 64 the three roots of which last equation, when substituted for p, q, and 7 in the formula x = v p + v r + vq, will give, by taking each term of the expression both in and -, all the four values of x. Or, in order to render this result more commodious in practice, by freeing it from fractions, let y=z. Then, by substitution and reduction, we shall have the corresponding equation z3+2az2+(a2 ·4c) z = b2, (4), the three roots of which are each, evidently, four times those of the former. Hence, using this instead of equation (3), and denoting its roots by r', '', ''", the last mentioned formula, taking each of its terms in and as before, will give the values of x, as in the above expressions. > 1 1 (3), Note.-If we were to take all the possible changes of the signs, in this case, which the terms of the assumed formula admit of, it would appear that x should have eight different values; but it is to be observed, that, according to the first part of the above investigation, the product VpX v q X v r = vh, or b; and consequently, that when b is positive, either all the three radicals must be taken in, or two in-, and one in +; and when b is negative, they must either be all- or two and one which considerations reduce the number of roots to four. 2 Note. If the three roots r', r", r'"', of the auxiliary cute equation 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. 0, to find the four roots 3. Given x4 25x2 + 60x 36 of the equation. 2 Here a 25, b = 60, and c = 36; Whence, by substituting these values for their equals in the cubic equation above given, we shall have 23 2 X 25 z +(25+4X36)z = 603, or z3 · 50z2+769 z = 3600: the three roots of which last equation, as found by trial, or by one of the former rules, are 9, 16, and 25, respectively; whence B • 1. Given 204 55x2 roots, or values of x. 2. Given x + 2x3 roots, or values of x. 4 3. Given x4. 8x3 +14x2 + 4x or values of x. 4. Given x4 or values of x. 7x2 X XC (−√9 · 6 3 - 4 - 5) √25) = 1/ ž 1 x = (−√9+ √16+ √25) = (−3+ 4+ 5) = +3 2 1 x = (+ √9√16+ √25)=(+34 +5)= +2 C OC 1⁄2 (+ √ 9 + √ 16 — √ 25) = ↓ ( + 3 + 4 5) +1 And consequently the four roots of the proposed equation are 1, 2, 3, and — 6. When b is negative, 2 EXAMPLES FOR PRACTICE. 30x + 504 = 0, to find the four Ans. 3, 7, 8x 12, to Ans. 1, 2, 8, to find the four roots, { 17x2 - 20x 6 = 0, to find the four roots, 2+√7, 2-17 4, and - 6. find the four 3, and - 2. Total 5. Given 204 3x2 4x3, to find the four roots, or 12 - 11 Ans. 13,--V-3. 13 302x+2000, to find the four .80955 values of x. 6. Given a1-19x3+132x2. 1 1 S + { roots, or values of x. Ans. +6.956377,±√(-9.3686). 7. Given x4 27x3 +162x2 + 356x 1200 0, to find 2.05608, - 3.00000 Ans. 13.15306, 14.79086 the four roots, or values of x. 8. Given x4 - 12x2 + 12x — 30, to find the four roots, .606018, 3.907378 or values of x. Ans. 22.858083, .443277. 9. Given x4 36x2+72x or values of x. Ans. 10. Given x4—12x3 + 47x2 roots, or values of x. 360, to find the four roots, $ 0.8729836, 1.2679494 4.7320506, 6.8729836. 72x + 36 = 0, to find the Ans. 1, 2, 3, and 6. 114x24x + 1 0, to find the + √ 197 14, 2 + √5 √ 197 — 14, 2 ·√5. 58x2-114x110, to find the 4 11. Given x + 24x3 roots, or values of x. Ans. 12. Given xa 6x3 roots, or values of x. Ans. 3+3 ± √(17±21 √3). 1 + 2 2 OF THE RESOLUTION OF EQUATIONS BY APPROXIMATION. EQUATIONS of the fifth power, and those of higher dimensions, cannot be resolved by any rule or algebraic 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 some of the usual methods of approximation; among which, that commonly employed is the following, 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. RULE. Find by trials, a number nearly equal to the root sought, which call r; and let z 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±z, and there will arise a new equation involving only z and known quantities. Reject all the terms of this equation in which z is of two or more dimensions; and the approximate value of z 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 abridged equation exhibiting the value of z; when a second correction of z will be obtained, which, being added 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 abovementioned 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 those in the several substitutions for the values of r.* EXAMPLES. 1. Given x3 + 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. r2 + 2rz + z2 90. XC r + z And by rejecting the terms z3, 3rza, and z2, as small in comparison with z, we shall have 23 + p2 +r+ 3r2z + 2rz +2, = 90; *It may here be observed, that if any of the roots of an equation be whole numbers, they may be determined by substituting 1, 2, 3, 4, &c., successively, both in plus and in minus, for the unknown quantity, till a result is obtained equal to that in the question; when those that are found to succeed, will be the roots required. Or, since the last term of any equation is always equal to the continued product of all its roots, the number of these trials may be generally diminished, by finding all the divisors of that term, and then substi tuting them both in plus and minus, as before, for the unknown quantity, when those that give the proper result will be the rational roots sought; but if none of them are found to succeed, it may be concluded that the equation cannot be resolved by this method; the roots, in that case, being either irrational or imaginary. |