Ex. To find the roots of the equation x3· 6x2+10x 8. First to take away the 2d term, its co-efficient being — 6, 2; put therefore x z + 2; then its 3d part is Here then a 2, b = 4, c},d=2. Theref./[d+(d2+c3)]=✨/[2+√(4—3%)]={/(2+√ 49°)= (2+3)= 1.57735, then the sum of these two is the value of z = 2. Hence x=x+2=4, one root of x in the eq. x3 6x2+10x=8. To find the two other roots, perform the division, &c. as in p. 261, thus: x —4) x3 — 6x2+10x - 8(x2 -2x+2=0 = it becomes x3+ y3 + 3xy (x + y) + a (x + y) = b. Again, suppose 3xy a; which substituted, the last equation becomes x + y = b. Now, from the square of this equation subtract four times the equation xy=ja, and there results 26 • 2x3y3 + y® = b2 + a3, the square root of which is x3 — y3 = √ (62+a). This being added to and taken from the equation x3+y3 b, gives = b + √ (b2 + + a3) = b + 2 v [(3b)2 + (}a)3], 2y3 = b 2y 3 · 2d + 2 v (d3 + c3)}. extracting the cube roots, we have x = d + √ (d2 + c3), and y = 3/d — v(d2 + c3); the sum of these two gives the first form of the root z above stated. And that the 2d form is equal to the first will be evident by reducing the two 2d quantities to the same denominator. When c is negative, and c3 greater than d2, the root appears in an imaginary form. VOL. I. 35 Hence x-2x=-2, or x2-2x+1=1, and x-1 = ±√−1;x=1+√ 1 or 1--1, the two - = other roots sought. Ans. x 5, or 1+/-3, or Ex. 4. Find the three roots of x3 + 6x = 20. 7x2+14x= 20. 1-3. OF SIMPLE INTEREST. As the interest of any sum, for any time, is directly proportional to the principal sum, and to the time; therefore the interest of 1 pound, for 1 year, being multiplied by any given principal sum, and by the time of its forbearance, in years and parts, will give its interest for that time. That is, if there be put r = the rate of interest of 1 pound per annum, t= the time it is lent for, and the amount or sum of principal and interest; then is prt the interest of the sum p, for the time t, and conseq. P+prt or p× (1+rt) = a, the amount for that time. From this expression, other theorems can easily be deduced, for finding any of the quantities above mentioned: which theorems, collected together, will be as follows: For Example. Required to find in what time any principal sum will double itself, at any rate of simple interest. In this case, we must use the first theorem, a = p + prt, in which the amount a must be made = 2p, or double the principal, that is, p + prt = 2p, or prt p, or rt = 1; and hence t= 1 Hence r being the interest of 17 for 1 year, it follows, that the doubling at simple interest, is equal to the quotient of any sum divided by its interest for 1 year. So, if the rate of interest be 5 per cent. then 100 520, is the time of doubling at that rate. Or the 4th theorem gives at once the principal sum, r = the rate of interest of 17 for 1 year, a = the whole amount of the principal and interest, there is another quantity employed in Compound Interest, viz. the ratio of the rate of interest, which is the amount of 11 for 1 time of payment, and which here let be denoted by R, viz. R= 1+r, the amount of 17 for 1 time. Then the particular amounts for the several times may be thus computed, viz. As 1l is to its amount for any time, so is any proposed principal sum, to its amount for the same time; that is, as 17 R P : PR, the 1st year's amount, PR2, the 2d year's amount, 17: R:: PR2: PR3, the 3d year's amount, and so on. Therefore, in general, pra is the amount for the t year, or t time of payment. Whence the following general theorems are deduced : a 1st, a = pr2 the amount ; 2d, p == the principal; R= Rt 3d, ==/ the ratio; 4th, t=log. of a-log. of the time. log. of R P From which, any one of the quantities may be found, when the rest are given. As to the whole interest, it is found by barely subtracting the principal p from the amount a. Example. Suppose it be required to find, in how many years any principal sum will double itself, at any proposed rate of compound interest. In this case the 4th theorem must be employed, making a=2p; and then it is So, if the rate of interest be 5 per cent. per annum; then R=1+05 = 1.05; and hence log. 2 log. 1.05 = •301030 = 14.2067 nearly; that is, any sum doubles itself in 14 years nearly, at the rate of 5 per cent. per annum compound interest. Hence, and from the like question in simple interest, above given, are deduced the times in which any sum doubles itself at several rates of interest, both simple and compound; viz. The following Table will very much facilitate calculations of compound interest on any sum, for any number of years, at various rates of interest. The Amounts of 17 in any Number of Years. 1 1.0300 1·0350 | 1·0400 | 1·0450 1.0500 1.0600 21.0609 1.0712 1.0816 1.0920 1.1025 1.1236 31.0927 1.1087 1.1249 1.1412 1.1576 1.1910 41.1255 1.1475 1.1699 1·1925 1-2155 | 1.2625 51.1593 1·1877 1.2167 1·2462 1.2763 1.3382 61-1948 1.2293 1·2653| 1·3023 1·3401 | 1.4185 7 1.2299 1.2723 1·3159 1-3609 1·4071 | 1·5036 8 1.2668 1.3168 1.3686 1.4221 1.4775 1.5939 9 1-3048 1.3629 1·4233 1.4861 10 1.3439 1.4106 1.4802 1.5530 1.6289 11 1.3842 1.4600 1.5895 1-6229 1.7103 12 1.4258 1.5111 1.6010 1.6959 1.7959 2.0122 13 1.4685 1.5640 1.6651 1-7722 1.8856 2.1329 141-5126 1.6187 1.7317 1.8519 1.9799 2.2609 151.5580 16 1.6047 1-7340 171.6528 1-7947 1.5513 1.6895 1.7909 1.8983 1.6753 1.8009 1.9353 2.0789 2.3966 1.87302-0224 2.1829 2.5404 1.9479 2-1134 2.2920 2.6928 181-7024 1.8575 2·0258 | 2.2085 2.4066 2.8543 19 1.7535 1.9225 2.1068 2.3079 2.5270 3.0256 20 1.8061 1.9828 2.1911 2-4117 | 2·6533 | 3.2071 The use of this Table, which contains all the powers, R', to the 20th power, or the amounts of 17, is chiefly to calculate the interest, or the amount of any principal sum, for any time, not more than 20 years. For example, let it be required to find, to how much 5231 will amount in 15 years, at the rate of 5 per cent. per annum compound interest. In the table, on the line 15, and in the column 5 per cent. Note 1. When the rate of interest is to be determined to any other time than a year; as suppose to a year, or a year, &c.: the rules are still the same; but then t will express that time, and R must be taken the amount for that time also. |