Their signs being contrary, the number a is not. properly speaking, their sum, but their difference. 400. When we make m=n, that is, if we suppose that the squares of the two required parts are equal, we have m=1; the first solution gives two equal parts and a, a result which is evident, whilst the second solution gives two infinite results (Art. 166), a These are proper results, according to the above enunciation, since that the quantities required must be infinitely great, with respect to their difference a, if we can suppose the ratio of their squares equal to unity. Now if a=18, m=25, and n=16; then substitut ing these values in the formula and a a√m' 1+√m' 1+√m" to the two parts re we shall find 10 and 8 equal quired, the same as in Ex. 2., which is a particular case of this general problem. Prob. 10. What two numbers are those, whose sum is to the greater as 10 to 7; and whose sum, multiplied by the less, produces 270? Ans. +21 and ±9. Prob. 11. What two numbers are those, whose difference is to the greater as 2 to 9, and the difference of whose squares is 128? Ans. 18 and ±14. Prob. 12. A mercer bought a piece of silk for 167. 4s.; and the number of shillings which he paid for. a yard was to the number of yards as 4 : 9. How many yards did he buy, and what was the price of a yard? Ans 27 yards, at 12s. per yard. Prob. 13. Find three numbers in the proportion of,, and; the sum of whose squares is 724. Ans. 12, 16, and ±18. Prob. 14. It is required to divide the number 14 into two such parts, that the quotient of the greater part, divided by the less, may be to the quotient of the less divided by the greater as 16:9. Ans. The parts are 8 and #6. Prob. 15. What two numbers are those whose difference is to the less, as 4 to 3; and their product, multiplied by the less, is equal to 504 ? 360. Ans. 14 and 6. Prob. 16. Find two numbers, which are in the proportion of 8 to 5, and whose product is equal to Ans. ±24, and ±15. Prob. 17. A person bought two pieces of linen, which, together, measured 36 yards. Each of them cost as many shillings per yard, as there were yards in the piece; and their whole prices were in the proportion of 4 to 1. What were the lengths of the pieces? Ans. 24. and 12 yards. Prob. 18. There is a number consisting of two digits, which being multiplied by the digit on the left hand, the product is 46; but if the sum of the digits be multiplied by the same digit, the product is only 10. Required the number. Ans. 23. Prob. 19. From two towns, C and D, which were at the distance of 396 miles, two persons, A and B, set out at the same time, and met each other, after travelling as many days as are equal to the difference of the number of miles they travelled per day; when it appears that A has travelled 216 miles. many miles did each travel per day? How Ans. A went 36, and B 30. Prob. 20. There are two numbers, whose sum is to the greater as 40 is to the less, and whose sum is to the less as 90 is to the greater. What are the numbers? Ans. 36, and 24. Prob. 21. There are two numbers, whose sum is to the less as 5 to 2; and whose difference, multiplied by the difference of their squares, is 135. Required the numbers. Ans. 9, and 6. Prob. 22. There are two numbers, which are in the proportion of 3 to 2; the difference of whose fourth powers is to the sum of their cubes as 26 to 7. Prob. 23. A number of boys set out to rob an Prob. 24. It is required to find two numbers such, that the product of the greater, and square of the less, may be equal to 36; and the product of the less, and square of the greater, may be 48. Ans. 4, and 3. Prob. 25. There are two numbers, which are in the proportion of 3 to 2 the difference of whose disser en f fourth powers is to the sum of their Required the numbers. Λ as 6 to Ans. 6, and 1. Prob. 26. Some gentlemen made an excursion, and every one took the same sum. Each gentleman had as many servants attending him as there were gentlemen; and the number of dollars which each had was double the number of all the servants; and the whole sum of money taken out was $3456. How many gentlemen were there? Ans. 12. Prob. 27. A detachment of soldiers from a regiment, being ordered to march on a particular service, each company furnished four times as many men as there were companies in the regiment; but those becoming insufficient, each company furnished 3 more men; when their number was found to be increased in the ratio of 17 to 16. How many companies were there in the regiment? Ans. 12. Prob. 28. A charitable person distributed a certain sum among some poor men and women, the numbers of whom were in the proportion of 4 to 5. Each man received one-third of as many shillings as there were women more than men. Now the men received all together 18s. more than the women. How many were there of each? 4 persons reluved Ans. 12 men, and 15 women. teach wom an had things as there were women more a m man men. Prob. 29. Bought two square carpets for 621. 1s.; for each of which I paid as many shillings per yard as there were yards in its side. Now had each of them cost as many shillings per yard as there were yards in the side of the other, I should have paid 17s. less. What was the size of each? Ans. One contained 31, and the other 64 square yards. Prob. 30. A and B carried 100 eggs between them to market, and each received the same sum. If A had carried as many as B, he would have received 18 pence for them; and if B had only taken as many as A, he would have received 8 pence. How many had each? Ans. A 40, and B 60. Prob. 31. The sum of two numbers is 5 (s), and their product 6(p): What is the sum of their 5th powers? Ans. 275 (s-5ps3+5p2s). CHAPTER X. ON QUADRATIC EQUATIONS. 401. Quadratic equations, as has been already observed, (Art. 388), are divided into pure and adfected. All pure equations of the second degree are comprehended in the formula x2=n, where n may be any number whatever, positive or negative, integral or fractional. And the value of x is obtained by extracting the square root of the number n; this value is double, for we have, (Art. 295), x=±√n, and in fact, (±/n)2=n. This may be otherwise explained, by observing, (Art. 106), that x2-n=(x+ √n). (x✓n)=0, and that any product consisting of two factors becomes nought, when there is no restriction in the equality to zero of that product, by making each of its factors equal to zero. We have, therefore, x=n; x=+√n, or x= ±√n. 402. Now, since the square root is taken on both sides of the equation, x2=n, in order to arrive at x=n; it is very natural to suppose that, x being the square root of x2, we should also affect x with the double sign; and, therefore, in resolving the equation x2 =n, we would write ±x=±√n; but by arranging these signs in every possible manner, namely: we would still have no more than the two first equations, that is, +x=±√n; for if we change the signs of the equations xn and -x+n, they become +x+n and +x=n, or x=±√n. 403. If, in the formula xn, n be negative, or which is the same thing, if we have x-n, where n is positive; then, x-n=±√nx√−1, and in fact (Vn) x (√1)3n X-1-n; therefore, the two roots of a pure equation are either both real, or both imaginary. 404. All adfected quadratic equations, after being properly reduced according to the rules pointed out in the reduction of simple equations, may be exhibited under the following general forms; namely x2+ · nxo, and x2+nx=n'; where n and n' may be any numbers whatever, positive or negative, integral or fractional, 405. The solution of adfected quadratic equations of the form xnxo, is attended with no difficulty; for the equation xnxo, being divided by x, becomes x+n=0, from which we find x=- 2. though we find only one value of x, according to this mode of solution, still there may be two values of x, which will satisfy the proposed equation. In the equation, x23x, for example, in which it is required to assign such a value of x, that x2 may become equal to 3x, this is done by supposing x=32 |