As 5151 :.123: 024 the correction, 7.100 add Therefore 7-124 nearly the root required. x= Note 5. The same rule also among other more difficult forms of equation, succeeds very well in what are called exponential ones, or those which have an unknown quantity in the exponent of the power; as in the following example: Ex. 4. To find the value of x in the exponental equation For more easily resolving such kinds of equations, it is convenient to take the logarithms of them, and then compute the terms by means of a table of logarithms. Thus, the logarithms of the two sides of the present equation are xx log. of x=2 the log. of 100. Then, by a few trials, it is soon perceived that the value of x is somewhere between the two numbers 3 and 4, and indeed nearly in the middle between them, but rather nearer the latter than the former. Taking therefore first x=3 5, and then =3.6, and working with the logarithms, the operation will be as follows: First supp. x=3.5 Log. of 3 5=0·544068 Second Supp. x=3·6 then 3-5 Xlog. 3.51.904238 then 3.6 Xlog. 3.6 = 2·002689 As 098451:1:: 002689: 0-00273 the correction 3 Ex. 5. To find the value of x in the equation x3+10x2 +5r260. Ans. x4.1179857. Ex. 6. To find the value of x in the equation x3-2x=50. Ans. 3-8648854. By repeating the operations with a larger table of logarithms, we find — 8.59728502354. Ed. Ex. 7. To find the value of x in the equation x3+2x223 x=70. Ans. x=5·13457. Ex. 8. To find the value of x in the equation x2 — 17x2+ 54x=350. Ans. x=14.95407. Ex. 9. To find the value of x in the equation x1 — 3x2 75x=10000. Ans. x= =10.2609. Ex. 10. To find the value of x in the equation 2x4 — 16x + 40x2-30x - 1. Ans. x= =1.284724. Ex. 11. To find the value of x in the equation x3+2x++ 3x3+4x2+5x=-54321. Ans. x 8.414455. Ex. 12. To find the value of x in the equation x = 123456789. Ans. 8-6400268. Ex. 13. Given 2x-7x3+11x2 -3x=11, to find . Ex. 14. To find the value of x in the equation (3.xa— 2/x+1)3 —(x3 —4x/x+31/x)*—56. Ans. 18.360877. To resolve Cubic Equations by Carden's Rule. THOUGH the foregoing general method, by the application of Double Position, be the readiest way, in real practice, of finding the roots in numbers of cubic equations, as well as of all the higher equations universally, we may here add the particular method commonly called Carden's Rule, for resolving cubic equations, in case any person should choose occasionally to employ that method. The form that a cubic equation must necessarily have to be resolved by this rule, is this, viz. 23+az=b, that is, wanting the second term, or the term of the 2d power z3. Therefore, after any cubic equation has been reduced down to its final usual form, x3+px2 + qx = r, freed from the co-efficient of its first term, it will then be necessary to take away the 2d term pa2; which is to be done in this manner : Takep, or of the co-efficient of the second term, and annex it with the contrary sign to another unknown letter z, thus z- p; then substitute this for x, the unknown letter in the original equation x2 + px2 + qxr, and there will result this reduced equation 23+ az=b. of the form proper for applying the following, or Carden's rule. Or take e=ļa, and d=1b, by which the reduced equation takes this form 23 +3ca2d. Then Then substitute the values of c and d in this form z = Vd+√(d3 +c3) + v/d−√(d3+c3), C or z = {/d+√(d2+c3) - Vd+√(d2+c3), and the value of the root z, of the reduced equation 23+ az=b, will be obtained. Lastly, take x=z-ip, which will give the value of x, the required root of the original equation x3+px2+qx=r, first proposed. One root of this equation being thus obtained, then depressing the original equation one degree lower, after the manner described p. 260 and 261, the other two roots of that equation will be obtained by means of the resulting quadratic equation. Note. When the co-efficient a, or c, is negative, and c3 is greater than d2, this is called the irreducible case, because then the solution cannot be generally obtained by this rule. Ex. To find the roots of the equation 3-6x2+10x = 8. First, to take away the 2d term, its co-efficient being — 6, its 3d part is-2; put therefore x = z+2, then x3=z3+6x2+12x+8 - 6x2 = -622-24z-24 Here then a=-2, b=4, c=-&.d=2. Theref. /d+(d3 +c3)=3/2+√(4-4)=√/2+√"= 1/2+3=1.57735 100 27 and d−√(d3 +c3)=3/2−√(4—4)=3/2 −√3 = 3/2-√3=0·42265 then the sum of these two is the value of z=2. Hence x = z+2=4, one root of x in the eq. x3-6x2+ I= 10x 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 Hence x2-2x=-2, or x2−2x+1=−1, and x-1=± √−1; x=1+ √-1 or 1-√-1, the two other sought. Ex. 2. To find the roots of x3-9x2+28x=30. Ans. x=3, or 3+/- 1. or 3-1. Ex. 3. To find the roots of x3-7x2+14x=20. Ans. x=5, or =1+√✓−3, or =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 a = the amount or sum of principal and interest; then = the interest of the sum p, for the time t, and conseq. p+prt or p×(1+rt)=a, the amount for that time. is prt From this expression, other theorems can easily be deduced, for finding any of the quantities above mentioned; which theorems collected together, will be as below: 1st, ap+prt, the amount, pr For Example. Let it be required to find, in what time any principal sum will double itself, at any rate of simple in terest. 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 = Here, Here, r being the interest of 11. 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 100-5-20, is the time of doubling at that rate. Or the 4th theorem gives at once BESIDES the quantities concerned in Simple Interest namely, p= the principal sum, r the rate or interest of 11. 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 11. for 1 time. Then the particular amounts for the several times may be thus computed, viz. As 17. is to its amount for any time, so is any proposed principal sum, to its amount for the same time; that is, as 1l.: Rp 11. R: PR2 and so on, PR, the 1st year's amount, : PR2, the 2d year's amount, PR3, the 3d year's amount, Therefore, in general, pR'=a is the amount for the t year, or t time of payment. Whence the following general theorems are deduced : |