required numbers refer to quantities which can be understood in opposite senses, as opposite directions, etc., an intelligible meaning can usually be given to a negative result. An imaginary result always implies inconsistent conditions. 23. The interpretation of a negative result is often facilitated by the following principle: If a given quadratic equation have a negative root, then the equation obtained from the given one by changing the sign of x has a positive root of the same absolute value. But equation (2) shows that r satisfies the equation ax2 - bx + c = 0, which is obtained from (1) by changing the sign of x. (1) (2) Pr. A man bought muslin for $3.00. If he had bought three yards more for the same money, each yard would have cost him 5 cents less. How many yards did he buy? Let x stand for the number of yards the man bought. Then 1 yard 300 cost cents. If he had bought x + 3 yards for the same money, each 300 yard would have cost х cents. x + 3 The root 12 satisfies the equation and also the conditions of the prob Equation (2) evidently corresponds to the problem: A man bought muslin for $3.00. If he had bought 3 yards less for the same money, each yard would have cost him 5 cents more. Notice that the intelligible result, 12, of the first statement has become 12 and is meaningless in the second statement. Attention is called to the remarks in Ch. XII., Art. 6. EXERCISES VII. 1. If 1 be added to the square of a number, the sum will be 50. What is the number? 2. If 5 be subtracted from a number, and 1 be added to the square of the remainder, the sum will be 10. What is the number? 3. One of two numbers exceeds 50 by as much as the other is less than 50, and their product is 2400. What are the numbers ? 4. The product of two consecutive integers exceeds the smaller by 17,424. What are the numbers? 5. If 27 be divided by a certain number, and the same number be divided by 3, the results will be equal. What is the number? 6. What number, added to its reciprocal, gives 2.9 ? 7. What number, subtracted from its reciprocal, gives n? Let n = 6.09. 8. If n be divided by a certain number, the result will be the same as if the number were subtracted from n. What is the number? Let n=4. 9. If the product of two numbers be 176, and their difference be 5, what are the numbers? 10. A certain number was to be added to, but by mistake was divided by the number. Nevertheless the correct result was obtained. What was the number? 11. If 100 marbles be so divided among a certain number of boys that each boy shall receive four times as many marbles as there are boys, how many boys are there? 12. The area of a rectangle, one of whose sides is 7 inches longer than the other, is 494 square inches. How long is each side? 13. The difference between the squares of two consecutive numbers is equal to three times the square of the less number. numbers? What are the 14. A merchant received $48 for a number of yards of cloth. If the number of dollars a yard be equal to three-sixteenths of the number of yards, how many yards did he sell? 15. In a company of 14 persons, men and women, the men spent $24 and the women $24. If each man spent $1 more than each woman, how many men and how many women were in the company? 16. A pupil was to add a certain number to 4, then to subtract the same number from 9, and finally to multiply the results. But he added the number to 9, then subtracted 4 from the number, and multiplied these results. Nevertheless he obtained the correct product. What was the number? 17. A man paid $80 for wine. If he had received 4 gallons less for the same money, he would have paid $1 more a gallon. How many gallons did he buy? 18. A man left $31,500 to be divided equally among his children. But since 3 of the children died, each remaining child received $3375 more. How many children survived? 19. Two bodies move from the vertex of a right angle along its sides at the rate of 12 feet and 16 feet a second respectively. After how many seconds will they be 90 feet apart? 20. A tank can be filled by two pipes, by the one in two hours less time than by the other. If both pipes be open 17 hours, the tank will be filled. How long does it take each pipe to fill the tank? 21. From a thread, whose length is equal to the perimeter of a square, 36 inches are cut off, and the remainder is equal in length to the perimeter of another square whose area is four-ninths of that of the first. What is the length of the thread? 22. A number of coins can be arranged in a square, each side containing 51 coins. If the same number of coins be arranged in two squares, the side of one square will contain 21 more coins than the side of the How many coins does the side of each of the latter squares con other. tain? 23. A farmer wished to receive $2.88 for a certain number of eggs. But he broke 6 eggs, and in order to receive the desired amount he increased the price of the remaining eggs by 2 cents a dozen. How many eggs had he originally? 24. Two bodies move toward each other from A and B respectively, and meet after 35 seconds. If it takes the one 24 seconds longer than the other to move from A to B, how long does it take each one to move that distance? 25. It takes a boat's crew 4 hours and 12 minutes to row 12 miles down a river with the current, and back again against the current. If the speed of the current be 3 miles an hour, at what rate can the crew row in still water? 26. A man paid $300 for a drove of sheep. By selling all but 10 of them at a profit of $2.50 each, he received the amount he paid for all the sheep. How many sheep did he buy? 27. Two men start at the same time to go from A to B, a distance of 36 miles. One goes 3 miles more an hour than the other, and arrives at B 1 hour earlier. At what rate does each man travel? 28. It took a number of men as many days to dig a ditch as there were If there had been 6 more men, the work would have been done in 8 days. How many men were there? men. 29. The front wheel of a carriage makes 6 revolutions more than the hind wheel in running 36 yards; if the circumference of each wheel were 1 yard longer, the front wheel would make but 3 revolutions more than the hind wheel in running the same distance. What is the circumference of each wheel? 30. Two men formed a partnership with a joint capital of $500. The first left his money in the business 5 months, and the second his money 2 months. Each realized $450, including invested capital. How much did each invest? 31. Two trains run toward each other from A and B respectively, and meet at a point which is 15 miles further from A than it is from B. After the trains meet, it takes the first train 23 hours to run to B, and the second train 3 hours to run to A. How far is A from B? 32. The perimeter of a rectangular lawn having around it a path of uniform width is 420 feet. The area of the lawn and path together exceeds twice the difference of their areas by 1200 square yards, and the width of the path is one-sixth of the shorter side of the lawn. Find the dimensions of lawn and path. 33. Water enters a forty-gallon cask through one pipe and is discharged through another. In 4 minutes one gallon more is discharged through the second pipe than enters through the first. The first pipe can fill the cask in 3 minutes less time than it takes the second to discharge 66 gallons. How long does it take the first pipe to fill the cask? 34. In a rectangle, whose sides are a and b inches respectively, a second rectangle is constructed. The sides of the inner rectangle are equally distant from the sides of the outer, and the area of the inner rectangle is one-nth of the remaining part of the outer. What are the lengths of the sides of the inner rectangle? Let a = 70, b = 521, n = 1. ... ... 35. It has been found by experiment that when an object is removed to a point 2, 3, 4, times its original distance from the source of light, its illumination is 22, 32, 42, times as feeble. A lamp and a candle are 4 feet apart. At what point on the line joining them will the illumination from the candle be equal to that from the lamp, if the light of the lamp be 9 times as intense as that of the candle? CHAPTER XXII. EQUATIONS OF HIGHER DEGREE THAN THE SECOND. We shall consider in this chapter a few higher equations which can be solved by means of quadratic equations. 1. A Binomial Equation is an equation of the form xna, wherein n is a positive integer. Certain binomial equations can be factored into linear and quadratic factors or factors which can be brought to quadratic form by proper substitutions. This example gives the three cube roots of 1, since 3 − 1 = 0 is equivalent to x3 1, or x = = 1. = Therefore the three cube roots of 1 are 1, + {√ − 3, − 1 − 1√−3. In general, the three cube roots of any number can be found by multiplying the principal cube root of the number in turn by the three algebraic cube roots of 1. E.g., 3/8=23/12, −1±√−3; the three cube roots of a are va, Va(−1±1√− 3), wherein 3⁄4a denotes the principal cube root of a. Ex. 2. Solve the equation x2 + 1 = 0. Factoring, (x2+1+x√2) (x2 + 1 − x √√/2)=0. This equation is equivalent to the two equations and x2+1+x√2 = 0, whence x = = {√2( − 1 ± √− 1); = {√2(1 ± √ − 1). Since the given equation is equivalent to x4 = 1, or x = −1, we conclude that the four fourth roots of - 1 are √2(1±√− 1), }√2(1 ±√− 1) |