To solve the equations x2+xy=15, xy-y3=2. Suppose and y=mx. x2+mx2=15, from the first equation, mx2- m2x2 = 2, from the second equation. Dividing one of these equations by the other, From this equation we can determine the values of m. 2 One of these values is 3' and putting this for m in the 78 2 equation x2+mx2=15, we get x2+ x2 = 15. From which we find x= ±3, 3 and then we can find y from one of the original equations. 259. The examples which we shall now give are intended as an exercise on the methods of solution explained in the four preceding articles. 1.23-y3=37. xy=16. 7. x2+xy+y2=39 3y2-5xy=25. xy+ y2=36. 12. x3-y3=152 XXI. ON PROBLEMS RESULTING IN QUADRATIC EQUATIONS. 260. THE method of stating problems resulting in Quadratic Equations does not require any general explanation. Some of the Examples which we shall give involve one unknown symbol, others involve two. Ex. (1) What number is that whose square exceeds the number by 42 ? And we find the values of x to be 7 or -6. Ex. (2) The sum of two numbers is 14 and the sum of their squares is 100. Find the numbers. Let x and y represent the numbers. Then and x+y=14, x2+ y2 = 100. Proceeding as in Art. 252 we find x=8 or 6, y=6 or 8. Hence the numbers are 8 and 6, EXAMPLES.-XCVIII. 1. What number is that whose half multiplied by its third part gives 864 ? 2. What is the number of which the seventh and eighth parts being multiplied together and the product divided by 3 the quotient is 298? 3 3. I take a certain number from 94. I then add the number to 94. I multiply the two results together and the result is 8512. What is the number? 4. What are the numbers whose product is 750 and the quotient of one by the other 3 ? 5. The sum of the squares of two numbers is 13001 and the difference of the same squares is 1449. Find the numbers. 6. The product of two numbers, one of which is as much above 21 as the other is below 21, is 377. Find the numbers: 7. The half, the third, the fourth and the fifth parts of a certain number being multiplied together the product is 6750. Find the number. 8. By what number must 11500 be divided, so that the quotient may be the same as the divisor, and the remainder 51 ? x2 = 11570-51 9. Find a number to which 20 being added, and from which 10 being subtracted, the square of the first result added to twice the square of the second result gives 17475. 10. The sum of two numbers is 26 and the sum of their squares is 436. Find the numbers. 11. The difference between two numbers is 17, and the sum of their squares is 325. What are the numbers? 12. What two numbers are they whose product is 255 and the sum of whose squares is 514 ? ✓ 13. Divide 16 into two parts such that their product added to the sum of their squares may be 208. S. A. 13 A 14. What number added to its square root gives as a result 1332? 15. What number exceeds its square root by 48 ? 16. What number exceeds its square root by 2550 ? 18. What two numbers are those whose sum multiplied by the greater is 204, and whose difference multiplied by the less is 35 ? large coeff 19. What two numbers are those whose difference is 5 and their sum multiplied by the greater 228 ? 20. Find three consecutive numbers whose product is equal to 3 times the middle number. 21. The difference between the squares of two consecutive numbers is 15. Find the numbers. 22. The sum of the squares of two consecutive numbers is 481. Find the numbers. 23. The sum of the squares of three consecutive numbers is 365. Find the numbers. NOTE. If I buy a apples for y pence, will represent the cost of an apple in pence. If I buy a sheep for z pounds, Z will represent the cost of a sheep in pounds. Ex. A boy bought a number of oranges for 16d. Had he bought 4 more for the same money, he would have paid one-third of a penny less for each orange. How many did he buy? Let a represent the number of oranges. 16 Then will represent the cost of an orange in pence. Hence x from which we find the values of x to be 12 or -16, Therefore he bought 12 oranges. 7 24. I buy a number of handkerchiefs for £3. Had I bought 3 more for the same money, they would have cost one shilling each less. How many did I buy? 25. A dealer bought a number of calves for £80. Had he bought 4 more for the same money, each calf would have cost £1 less. How many did he buy? 26. A man bought some pieces of cloth for £33. 15s., which he sold again for £2. 8s. the piece, and gained as much as one piece cost him. What did he give for each piece? 27. A merchant bought some pieces of silk for £180. Had he bought 3 pieces more, he would have paid £3 less for each piece. How many did he buy? 28. For a journey of 108 miles 6 hours less would have How many 108 -6); x+3)= 29. A grazier bought as many sheep as cost him £60. Out of these he kept 15, and selling the remainder for £54, gained 2 shillings a head by then. How many sheep did he buy? 1200-42 1080 15 30. A cistern can be filled by two pipes running together in 2 hours, 55 minutes. The larger pipe by itself will fill it sooner than the smaller by 2 hours. What time will each pipe take separately to fill it? 31. The length of a rectangular field exceeds its breadth 32. A certain number consists of two digits. The left- 33. A ladder, whose foot rests in a given position, just А 10 2268 a 13-2 |