Problems Involving Quadratics. 310. Problems that involve quadratic equations apparently have two solutions, since a quadratic equation has two roots. If both roots of the quadratic equation are positive integers, they will, generally, both be admissible solutions. Fractional and negative roots will in some problems give admissible solutions; in other problems they will not. No difficulty will be found in selecting the result which belongs to the particular problem we are solving. Sometimes, by a change in the statement of the problem, we may form a new problem which corresponds to the result that was inapplicable to the original problem. Imaginary roots indicate that the problem is impossible. Here, as in simple equations, x stands for an unknown number. 1. The sum of the squares of two consecutive numbers is 481. Find the numbers. The positive root 15 gives for the numbers, 15 and 16. The negative root - 16 is inapplicable to the problem, as consecutive numbers are understood to be integers which follow one another in the common scale, 1, 2, 3, 4 ..... 2. A pedler bought a number of knives for $2.40. Had he bought 4 more for the same money, he would have paid 3 cents less for each. How many knives did he buy, and what did he pay for each? Let Then, x= number of knives he bought. 240 х = number of cents he paid for each. He bought 16 knives, therefore, and paid 240, or 15 cents for each. If the problem is changed so as to read: A pedler bought a number of knives for $2.40; if he had bought 4 less for the same money, he would have paid 3 cents more for each, the equation will be This second problem is therefore the one which the negative answer of the first problem suggests. 3. What is the price of eggs per dozen when 2 more in a shilling's worth lowers the price 1 penny per dozen? And, if 16 eggs cost a shilling, 1 dozen will cost 9 pence. Therefore, the price of the eggs is 9 pence per dozen. If the problem is changed so as to read: What is the price of eggs per dozen when two less in a shilling's worth raises the price 1 penny per dozen? the equation will be Hence, the number 18, which had a negative sign and was inapplicable in the original problem, is here the true result. EXERCISE 120. 1. The sum of two squares of two consecutive integers is 761. Find the numbers. 2. The sum of the squares of two consecutive numbers exceeds the product of the numbers by 13. Find the numbers. 3. The square of the sum of two consecutive even numbers exceeds the sum of their squares by 336. Find the numbers. 4. Twice the product of two consecutive numbers exceeds the sum of the numbers by 49. Find the numbers. 5. The sum of the squares of three consecutive numbers is 110. Find the numbers. 6. The difference of the cubes of two consecutive odd numbers is 602. Find the numbers. 7. The length of a rectangular field exceeds its breadth by 2 rods. If the length and breadth of the field were each increased by 4 rods, the area would be 80 square rods. Find the dimensions of the field. 8. The area of a square may be doubled by increasing its length by 10 feet and its breadth by 3 feet. Find the length of its side. 9. A rectangular grass plot 12 yards long and 9 yards wide has a path around it. The area of the path is of the area of the plot. Find the width of the path. 10. The perimeter of a rectangular field is 60 rods. Its area is 200 square rods. Find its dimensions. 11. The length of a rectangular plot is 10 rods more than twice its width, and the length of a diagonal of the plot is 25 rods. What are the dimensions of the plot? 12. The denominator of a certain fraction exceeds the numerator by 3. If both numerator and denominator are increased by 4, the fraction will be increased by . Find the fraction. 13. The numerator of a fraction exceeds twice the denominator by 1. If the numerator is decreased by 3, and the denominator increased by 3, the resulting fraction will be the reciprocal of the given fraction. Find the fraction. 14. A farmer sold a number of sheep for $120. If he had sold 5 less for the same money, he would have received $2 more a sheep. How much did he receive a sheep? State the problem to which the negative solution applies. 15. A merchant sold a certain number of yards of silk for $40.50. If he had sold 9 yards more for the same money, he would have received 75 cents less per yard. How many yards did he sell? 16. A man bought a number of geese for $27. He sold all but two for $25, thus gaining 25 cents on each goose sold. How many geese did he buy? 17. A man agrees to do a piece of work for $48. It takes him 4 days longer than he expected, and he finds that he has earned $1 less per day than he expected. In how many days did he expect to do the work? 18. Find the price of eggs per dozen when 10 more in one dollar's worth lowers the price 4 cents a dozen. 19. A man sold a horse for $171, and gained as many per cent on the sale as the horse cost dollars. How much did the horse cost? 20. A drover bought a certain number of sheep for $160. He kept four, and sold the remainder for $10.60 per head, and made on his investment as many per cent as he paid dollars for each sheep bought. How many sheep did he buy? 21. Two pipes running together can fill a cistern in 5§ hours. The larger pipe will fill the cistern in 4 hours less time than the smaller. How long will it take each pipe running alone to fill the cistern? 22. A and B can do a piece of work together in 18 days, and it takes B 15 days longer to do it alone than it does A. In how many days can each do it alone? 23. A boat's crew row 4 miles down a river and back again in 1 hour and 30 minutes. Their rate in still water is 2 miles an hour faster than twice the rate of the current. Find the rate of the crew and the rate of the current. 24. A number is formed by two digits. The units' digit is 2 more than the square of half the tens' digit, and if 18 is added to the number, the order of the digits will be reversed. Find the number. 25. A circular grass plot is surrounded by a path of a uniform width of 3 feet. The area of the path is the area of the plot. Find the radius of the plot. 26. If a carriage wheel 11 feet round took of a second less to revolve, the rate of the carriage would be five miles more per hour. At what rate is the carriage traveling? |