21. A certain farm is a rectangle, whose length is twice its breadth. If it should be enlarged 20 rods in length, and 24 rods in breadth, its area would be doubled. Of how many acres does the farm consist? 22. A square court-yard has a gravel-walk around it. The side of the court lacks one yard of being 6 times the breadth of the walk, and the number of square yards in the walk exceeds the number of yards in the perimeter of the court by 340. Find the area of the court and the width of the walk. 23. A merchant bought 54 bushels of wheat, and a certain quantity of barley. For the former he gave half as many dimes per bushel as there were bushels of barley, and for the latter 40 cents a bushel less. He sold the mixture at $1 per bushel, and lost $57.60 by the operation. Required the quantity of barley, and its price per bushel. 24. A certain number consists of two digits, the lefthand digit being twice the right-hand. If the digits are inverted, the product of the number thus formed, increased by 11, and the original number, is 4956. Find the number. 25. 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 separately take to fill it? 26. A and B gained by trade $1800. A's money was in the firm 12 months, and he received for his principal and gain $2600. B's money, which was $3000, was in the firm 16 months. How much money did A put into the firm? 27. My gross income is $1000. After deducting a percentage for income tax, and then a percentage, less by one than that of the income tax, from the remainder, the income is reduced to $912. Find the rate per cent of the income tax. 28. A man travelled 102 miles. If he had gone 3 miles more an hour, he would have performed the journey in 5 hours less time. How many miles an hour did he go? 29. The number of square inches in the surface of a cubical block exceeds the number of inches in the sum of its edges by 210. What is its volume? 30. A man has two square lots of unequal size, containing together 15,025 square feet. If the lots were contiguous, it would require 530 feet of fence to embrace them in a single enclosure of six sides. Required the area of each lot. 31. A set out from C towards D at the rate of 3 miles an hour. After he had gone 28 miles, B set out from D towards C, and went every hours of the entire distance; and after he had travelled as many hours as he went miles in an hour, he met A. Required the distance from C to D. 32. A courier proceeds from P to Qin 14 hours. A second courier starts at the same time from a place 10 miles behind P, and arrives at Q at the same time as the first courier. The second courier finds that he takes half an hour less than the first to accomplish 20 miles. Find the distance from P to Q. 33. A person bought a number of $20 mining-shares when they were at a certain rate per cent discount for $1500; and afterwards, when they were at the same rate per cent premium, sold them all but 60 for $1000. How many did he buy, and what did he give for each of them? XXIII. EQUATIONS IN THE QUADRATIC FORM. 269. An equation is in the quadratic form when it is expressed in three terms, two of which contain the unknown quantity; and of these two, one has an exponent twice as great as the other; as, x - 6x = 16; 3 = 72; (x2 - 1)2 + 3(x2 - 1) = 18; etc. 270. The rules for the solution of quadratics are applicable to equations having the same form. 1. Solve the equation x2 - 6x = 16. Completing the square, x-6x+9=16+ 9 = 25. Extracting the square root, x-3=±5. Whence, x=3±5 = 8 or - 2. Extracting the cube root, x=2 or -2, Ans. Note. There are also four imaginary roots, which may be obtained by the method explained in Art. 282. 2. Solve the equation 2x+3√x=27. Since x is the same as 22, this is in the quadratic form. Multiplying by 8, and adding 32 or 9 to both members, 16x+24√x+9=216+9 = 225. Extracting the square root, 4x 1√x + 3 = ± 15. Or, 4√x=-3±15 = 12 or 18. 3. Solve the equation 16x – 22 x = 3. Multiplying by 16, and adding 112 to both members, 162x – 16 × 22x + 121 = 48 + 121 = 169. Extracting the square root, 16x ̄ - 11 = ± 13. P Ans. , Note. In solving equations of the form x = a, first extract the root corresponding to the numerator, and afterwards raise to the power corresponding to the denominator. Particular attention should be paid to the algebraic signs; see Arts 192 and 201. 271. An equation may be solved with reference to an expression, by regarding it as a single quantity. 1. Solve the equation (x - 5)3 - 3(x-5) = 40. Regarding x-5 as a single quantity, we complete the square in the usual way. Multiplying by 4, and adding 9 to both members, 3 4(x - 5) – 12 (x - 5) +9 9 = 160 + 9 = 169. Extracting the square root, |