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17. Three merchants, A, B, C, on comparing their gains find, that aniong them all they have gained 14441. ; and that B's gain added to the square root of a's made 9201 ; but if added to the square root of c's it made 912.
What were their several gains ?
Ans. A 400, B 900, c 144. 18. To find three numbers in arithmetical progression, so that the sum of their squares shall be 93 ; also it the first be multiplied by 3, the second by 4, and the third by 5, the sum of the products may be 66.
Ans 2, 5, 8. 19. To find four numbers such, that the first may be to the second as the third to the fourth ; and that the first may be to the fourth as I to 5; also the second to the third as 5 to 9; and the sum of the second and fourth may be 20.
Ans. 3, 5, 9, 15. 20. To find two numbers such that their product added to their sum may make 47, and their sum taken from the sum of their squares may leave 62.
Ans. 5, and 7.
RESOLUTION OF CUBIC AND HIGHER
EQUATIONS. A Cubic Equation, or Equation of the 3d degree or power, is one that contains the third power of the unkuown quantity. As x3 axa + bx = c.
A Biquadratic, or Double Quadratic, is an equation that contains the 4th Power of the unknown quantity :
As x*- Qrs + bx? -cx = d. An Equation of the 5th Power or Degree, is one that con. tains the 5th power of the unknown quantity :
As 25 - ar* + 6.303 cm + dx = e. And so on, for all other higher powers. Where it is to be noted, however that all the powers, or terms in the equation, are supposed to be freed from surds or fractional exponents
There are many particular and prolix rules usually given for the solution of some of the above-mentioned powers or equations. Bu: they may be all readily solved by the following easy rule of Double Position, sometimes called Trial-and.error.
1. Find, by trial, two numbers, as near the true root as you can, and substitute them separately in the given equation, instead of the unknown quantily ; and find how much the Voz. I. LI
terms collected together, according to their signs + or -, differ from the absolute known term of the equation, marking whether these errors are in excess or defect.
2. Multiply the difference of the iwo numbers, found or taken by irial, by either of the errors, and divide the product by the difference of the errors, when they are alike, but by their sum when they are unlike. Or say, As the difference or sum of the errors, is to the difference of the two numbers, so is either error to the correction of its supposed number.
3. Add the quotient, last found, to the number belonging to that error, when its supposed number is too little, but subtract it when too great, and the result will give the true root nearly
4. Take this root and the nearest of the two former, or any other that may be found nearer; and, by proceeding in like manner as above, a root will be had still nearer than before. And so on to any degree of exactness required.
Note 1. It is best to employ always two assumed numbers that shall differ from each other only by unity in the last figure on the right hand; because then the difference, or multiplier, is only 1. It is also best to use always the least error in the above operation.
Note 2. It will be convenient also to begin with a single figure at first, trying several single figures till there be found the two nearest the truth, the one too little, and the other too great ; and in working with them, find only one more figure. Then substitute this corrected result in the equation, for the unknown letter, and if the result prove too little, substitute also the number next greater for the second supposition; but contrarywise, if the former prove too great, then take the next less number for the second supposition : and in working with the second pair of errors, continue the quotient only so far as to have the corrected number to four places of figures. Then repeat the same process again with this last corrected number, and the next greater or less, as the case may require, carrying the third corrected number to eight figures; because each new operation commonly doubles the puniber of true figures. And thus proceed to any extent that may be wanted.
Ex. 1. To find the root of the cubic equation #4 + 284 x = 100, or the value of x in it,
Here it is soon found that Again, suppose 4.2 and 4:3;
lies between 4 and 5. As and repeat the work as fol. sunie therefore these two num lows : bers, and the operation will be as follows: Ist Sup. 2d Sup. 1st Sup.
2d Sup. 4 5 4.2
4.3 25 17.64
18 49 64
125 74 088
Again, suppose 4.264, and 4.265, and work as follows: 4.264
4.265 18 181696
18.190225 77 5267 52
99 972448 100
gives x very nearly = 4.2644299
The work of the example above might have been much shortened, by the use of the Table of Powers in the Arithmetic, which would have given iwo or three figures by in. spection. But the example has been worked out so particularly as it is, the better to show the method.
Ex.2. To find the root of the equation x3 15 x2 + 63 x 50, or the value of x in it. Here it soon appears that x is very little above l.
Suppose therefore 1.0 and 1.1,| Again, suppose the two numand work as follows:
bers 1 03 and 1.02 &c. as
1.02 630 . 63.r 69.3 64.89 637 64.25 - 1572 -18.15
-15 9135 - 1572 - 15.6060 1 1 331 1:09 2727 x 3
Note 3. Every equation has as many roots as it contains dimensions, or as there are units in the index of its highest power. That is, a simple equation has only one value of the root; but a quadratic equation has two values or roots, a cubic equation has three roots, a biquadratic equation bas four routs, and so on.
And when one of the roots of an equation has been found by approximation, as above, the rest may be found as follows. Take, for a dividend, the given equation, with the known term transposed, with its sign changed, to the unknown side of the equation ; and, for a divisor, take x minus the root just found. Divide the said dividend by the divisor, and the quotient will be the equation depressed a degree lower than the given one.
Find a root of this new equation by approximation, as before, or otherwise, and it will be a second root of the original equation. Then, by means of this root, depress the second equation one degree lower, and from thence find a third root, and so on, till the equation be reduced to a quadratic ; then the two roots of this being found, by the method of completing the square, they will make up the remuinder of the roots. Thus in the foregoing equation, having found one root to be 1•02804, connect it by minus with
for a divisor, and the equation for a dividend, &c. as follows: 1.02804) 3 - 153% + 63x
50 (2.2 13 97196.3 + 48.63627
Then the two roots of this quadratic equation, or -12 – 13.97196 * = 48 63627, by completing the square, are 6.57653 and 7.39543, which are also the other two roots of the given cubic equation. So that all the three roots of that equation, viz. 23 15 x3 + 63x = 50. are 1.02804
and the sum of all the roots is found to be and 6-57653 and 7.39543
15, being equal to the co-efficient of the 2d term of the equation, which the sum of
"he roots always ought to be, when they are sum 15 00000
Note 4. It is also a particular advantage of the foregoing rule, that it is not necessary to prepare the equation, as for other rules, by reducing it to the usual final form and state of equations. Because the rule may be applied at once to an unreduced equation, though it be ever so much embarrassed by surd and compound quantities. As in the following example:
Ex. 3. Let it be required to find the root x of the equation ✓ 14402 (t? + 20): + V 196x* - (1% + 24,2 > 114, or the value of x in it.
By a few trials, it is soon found that the value of x is but little above 7. Suppose therefore first that x is = 7, and then x = 8. First, when x = 7.
Second, when x = 8,
Suppose again x = 7.2 , and then, because it turns out too great suppose also = 7.1, &c, as follows: