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2. To find two numbers such, that the less may be to the greater as the greater is to 12, and that the sum of their squares may be 45.
Ans. S and 6. 3. What two numbers are those, whose difference is 2, and the difference of their cubes 98 ?
Ans. 3 and 5. 4. What two numbers are those whose sum is 6, and the sum of their cubes 72 ?
Ans. 2 and 4. 5. What two numbers are those, whose product is 20, and the difference of their cubes 61 ? Ans. 4 and 5.
6. To divide the number 11 into two such parts, that the product of their squares may be 784.
Ans. 4 and 7. 7. To divide the number 5 into two such parts, that the sum of their alternate quotients may be 45, that is of the two quotients of each part divided by the other.
Ans. 1 and 4. 8. To divide 12 into two such parts, that their product may be equal to 8 times their difference. Ans. 4 and 8.
9. To divide the number 10 into two such parts, that the square of 4 times the less part, may be 112 more than the square of 2 times the greater.
Ans. 4 and 6. 10. To find two numbers such, that the sum of their squares may be 89, and their sum multiplied by the greater may produce 104.
Ans. 5 and 8. 11. What number is that, which being divided by the product of its two digits, the quotient is 5%; but when 9 is subtracted from it, there remains a number having the same digits inverted ?
Ans. 32. 12. To divide 20 into three parts such, that the continual product of all three may be 270, and that the difference of the first and second may be 2 less than the difference of the second and third.
Ans. 5, 6, 9. 13. To find three numbers in arithmetical progression, such that the sum of their squares may be 56, and the sum arising by adding together 3 times the first and 2 times the second and 3 times the third, may amount to 28.
Ans. 2, 4,6. 14. To divide the number 18 into three such parts, that their squares may have equal differences, and that the sum of those squares may be 75.
Ans. 1, 5, 7, 15. To find three numbers having equal differences, so that their sum may be 12, and the sum of their fourth powers 962.
Ans. 3, 4,
16. To find three numbers having equal differences, and such that the square of the least added to the product of the two greater may make 28, but the square of the greatest added to the product of the two less may make 44.
Ans. 2, 4, 6. 17. Three merchants, A, B, C, on comparing their gains find, that among 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 if 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 1 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
A CUBIC Equation, or Equation of the 3d degree or power,
is one that contains the third power of the unknown quantity. As 43 -- a.x? + bx = c.
A Biquadratic, or Double Quadratic, is an equation that contains the 4th power of the unknown quantity :
As x4 - ax3 + bx? -0% = d. An Equation of the 5th Power or Degree, is one that contains the 5th power of the unknown quantity :
As x5 - 0x4 + 6x3 – cx? + 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. But 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 quantity; and find how much the 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 two numbers, found or taken by trial, 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 number of true figures. And thus proceed to any extent that may be wanted.
Ex. 1. To find the root of the cubic equation .213 +*+ x = 100, or the value of x in it.
Here it is soon found that Again, suppose 4.2 and 4:3, x lies between 4 and 5. As- and repeat the work as fol. sume therefore these two num- lows: bers, and the operation will be as follows: 1st Sup
2d Sup. 1st Sup.
Again, suppose 4.264, and 4.265, and work as follows: 4.264
gives * 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 two or three figures by inspection. 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 r3 – 15x2 + 63.3 = 50, or the value of x in it.
Here it soon appears that x is very little above 1. Suppose therefore 1.0 and 1.1, Again, suppose the two numand work as follows: bers 1.03 and 1.02, &c, as
errors +2:431 +069227 errors - 284792 3.481 sum of the errors. •284792 As 3:481:1::1:03 correct.
-0019555 Hence r= 1.03 nearly. This taken from 1.03
leaves I nearly
Note 3. Every equation las 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 has four roots, 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.