that is, x2 ·42x + 3 ( = ±x√ 6 + 3x 5: which, solved, gives 15) =± The Resolution of LITERAL EQUATIONS, wherein the given, and the unknown quantity, are alike affected. Equations of this kind, in which the given and the unknown quantities can be substituted, alternately, for each other, without producing a new equation, are always capable of being reduced to others of lower dimensions. In order to such a reduction, let the equation, if it be of an even dimension, be first divided by the equal powers of its two quantities in the middle term: then assume a new equation, by putting some quantity (or letter) equal to the sum of the two quotients that arise by dividing those quantities one by the other, alternately; by means of which equation, let the said quantities be exterminated; whence a numeral equation will emerge, of half the dimensions with the given literal one. But, if the equation propounded be of an odd dimension, let it be, first, divided by the sum of its two quantities, so will it become of an even dimension, and its resolution will therefore depend upon the preceding rule. Exam. 1. Let there be given the equation x2 - 4аx3 + 5a2x2- 4a3x + a4 = 0. - O, by joining the corresponding terms;) and by mak fore, by substituting these values, our equation becomes -4x+5=0, or z2-4x=-3; whence z=3. Bat %, we have x2-zax-a2; and conse Exam. 2. Let there be given x5+4ax1-12a2x3-12a3x2 +4a*x+a=0. In this case, we must first divide by x+a, and the quotient will come out + 3ax3-15a2x2 + зα3x + a2=0: whence, by proceeding as in the former example, we have aa хх Xx + + 3 x + a aa 15=0, or z2 — 2 + 3≈—15 a x Exam. 3. Suppose there to be given 7x-26ax3 26a5x+7a=0. - 2 = = x2 a2 a2 + ; and multiplying again we likewise have 3 2%= X3 3%= a3 X3 ed above, 26 X x2 3% 2 = 0, or 723 26%2 our equation becomes 7 x Where, trying the divisors of the last term, which are 1, 2, 4, 13, &c. the third is found to answer; ≈, consequently, being 4. Exam. 4. Wherein let there be given 2x7-13a2x5 —— 1 Sax2+2a7=0. Here dividing, first, by x+a, the quotient will be 2x6 -2аx5-11a2x2+11a3x3-11a*x2—2α3x+2a = 0; which, X3 a3 divided again by a3x3, gives 2× + a 2 X x2 11 X + +110, that is, 2 x z3-3- 2 x a 22—2—11x+11=0, or 2z3. 2x2-17% +15=0 (vid. p. 119:) whence == 3. A literal equation may be made to correspond with a numeral one, by substituting an unit in the room of the given quantity (or letter:) and equations that do not seem, at first, to belong to the preceding class, may sometimes be reduced to such, by a proper substitution; that is, by putting the quotient of the first term divided by the last, equal to some new unknown quantity (or letter) raised to the power expressing the dimension of the equation. Thus, if the equation given be 2x1+24x3—315x2 2X4 +216x+162=0; by putting 162=y', we have x=3y; whence, after substitution, the given equation becomes 162y+648y3-2835y+648y+162-0: which now answers to the rule. and may be reduced down to 2y+8y3 —35y2+8y+2= Of the Resolution of Equations by Approximation and converging Series. The methods hitherto given, for finding the roots of equations, are either very troublesome and laborious, or else confined to particular cases; but that by converging series, which we are here going to explain, is universal, extending to all kinds of equations; and, though not accurately true, gives the value sought, with little trouble, to a very great degree of exactness. When an equation is proposed to be solved by this method, the root thereof must, first of all, be nearly estimated (which, from the nature of the problem and a few trials, may, in most cases, be very easily done;) and some letter, or unknown quantity (as z) must be assumed, to express the difference between that value, which we will call r, and the true value (x;) then, instead of x, in the given equation, you are to substitute its equal r±, and there will emerge a new equation, affected only with ≈ and known quantities; wherein all the terms having two, or more dimensions of z, may be rejected, as inconsiderable in respect of the rest; which being done, the value of will be found, by the resolution of a simple equation; from whence that of x (= r±z) will also be known. But, if this value should not be thought sufficiently near the truth, the operation may be repeated, by substituting the said value instead of r, in the equation exhibiting the value of ; which will give a second correction for the value of x. As an example hereof, let the equation x3 + 10x2 +50x2600, be proposed: thence, since it appears that x must, in this case, be somewhat greater than 10, let r be put 10, and r+= x; which value being substituted for x, in the given equation, we have gu3 + 3p2x + 3rz2 + z3 + 10r2 + 20rz + 10z2 + 50r +50%=2600: this, by rejecting all the terms wherein two or more dimensions of duced to 3 + 3r2 + 10r2 + = 2600; whence comes out are concerned, is re20rz + 50r + 50% = = 0.18, nearly which, added to 10 (= r,) gives 10,18 for the value of x. But, in order to repeat the operation, let this value be substituted for r, in the last equation, and you will have ,0005347; which, added to 10,18, gives 10,1794653, for the value of x, a second time corrected. And, if this last value be, again, substituted for r, you will have a third correction of a; from whence a fourth may, in like manner, be found; and so on, until you arrive to what degree of exactness you please. But, in order to get the general equation from whence these successive corrections are derived, with as little trouble as possible, you may neglect all these terms, which, in substituting for x and its powers, would rise to two or more dimensions of the converging quantity: for, they being, by the rule, to be omitted, it is better entirely to exclude them, than to take them in, and afterwards reject them. = Thus, in the equation x + x2 + x = 90, let r + z be put = x; and then, by omitting all the powers of ≈ above the first, we shall have r2 + 2rz x2, and 3 + 3r2 = x3, nearly; which, substituted above, give 13 + 37·2% + r2 + 2rz+r+x=90; whence z is found = 32+2r+1 Therefore, if r be now taken equal to 4 (which, it is easy to perceive, is nearly the true value of x) we shall have 90-64-16-4 6 57) =0.10, &c. which, added to 4, gives 4.1, for the value of x, once corrected: and, if this value of x be now substituted for r, we shall have≈ 90—p3—p2—r = - 3r2 + 2r+1) =,00283; which, added to 4.1, gives 4.10283, for the value of x, a second time corrected. In the same manner, a general theorem may be derived, for equations of any number of dimensions. Let ax" + Bant Ca2+ den 3 + lan, ga. Q, be such an equation, where n, a, b, c, d, &c. represent any given quantities, positive, or negative; then, putting r + x = x, we have, by the theorem in p. 41. |