18. Reduce += √1 + √ 2 2x0°3 + 202. x 4 Ans. x 12, or 4. 19. Reduce 6+4x-12x2100 x 2. 25. Reduce 6+1184 = 58 x = ±80 ± 26. Reduce x + 3 − √ x + 3 = 2. √x+3=2. +1-3241 -3241.781 Ans. x 13, or 2. 27. Reduce 2 - √x2 + x-525-x. NOTE. By transposing x and subtracting 5 from each member, make the expression without the radical in the first member like that under the radical; then complete the square, &c. 28. Reduce x2 − 2 x + 3 No√√2x2 — 6 x — 11 = x + 33. 32. Reduce x + (x2 — x)2 = x2 + 5112. PROBLEMS PRODUCING AFFECTED QUADRATIC EQUATIONS WITH BUT ONE UNKNOWN QUANTITY. 182. Though the numerical negative values obtained in solving the following Problems satisfy the equations formed in accordance with the given' conditions, they are practically inadmissible, and are therefore not given in the answers. 1. Divide 40 into two parts such that the sum of their squares shall be 1042. 2. Divide 20 into two parts such that their product will be 992. Ans. 9 and 10. 3. The ages of two brothers are such that the age of the elder plus the square root of the age of the younger is 22 years, and the sum of their ages is 34 years. What is the age of each? Ans. Elder, 18; younger, 16. NOTE. --- The other answers found by reducing the equation, viz. 25 and 9, satisfy the conditions of the equation only upon considering 9: 3. To make the problem correspond to these answers, the word "plus" must be changed to "minus." 4. A merchant had two pieces of cloth measuring together 96 yards. The square of the number of yards in the longer is equal to one hundred times the number of yards in the shorter. How many yards are there in each piece? Ans. 60 and 36. 5. Find two numbers whose difference is 3, and the sum of whose squares is 117. Ans. 9 and 6. 6. A merchant having sold a piece of cloth that cost him $42, found that if the price for which he sold it were multiplied by his loss, the product would be equal to the cube of the loss. What was his loss? NOTE. If the word "loss" were changed to gain, the other answer, — 7, or as it would then become, +7, would be correct. Ans. $6. 7. Find two numbers whose difference is 5, and product 176. Ans. 11 and 16. 8. There is a square piece of land whose perimeter in rods is 96 less than the number of square rods in the field. What is the length of one side? Ans. 12 rods. 9. Find two numbers whose sum is 8, and the sum of whose cubes is 152. 10. A man bought a number of sheep for $240, and sold them again for $6.75 apiece, gaining by the bargain as much as 5 sheep cost him. How many sheep did he buy? Ans. 40. H 11. Find two numbers whose difference is 4, and the sum of whose fourth powers is 1312. NOTE. Let x 2 and 2 be the numbers. Ans. 2 and 6. 12. A man sold a horse for $312.50, and gained one tenth as much per cent as the horse cost him. How much did the horse cost him? Ans. $250. 13. The difference of two numbers is 5, minus the square root of the greater is 7. numbers? and the less What are the Ans. 11 and 16. 14. A and B started together for a place 300 miles distant. A arrived at the place 7 hours and 30 minutes before B, who travelled 2 miles less per hour than A. How many miles did each travel per hour? Ans. A, 10; B, 8 miles. 15. A gentleman distributed among some boys $15; if he had commenced by giving each 10 cents more, 5 of the boys would have received nothing. How many boys were there? Ans. 30. 16. Find two numbers whose sum is a, and product b. Ans. 2 a = √ a2 46 2 17. A merchant bought a piece of cloth for $45, and sold it for 15 cents more per yard than he paid. Though he gave away 5 yards, he gained $4.50 on the piece. How many yards did he buy, and at what price per yard? Ans. 60 yards, at 75 cents per yard. 18. A certain number consists of two figures whose sum is 12; and the product of the two figures plus 16 is equal to the number expressed by the figures in inverse order. What is the number? Ans. 84. 19. From a cask containing 60 gallons of pure wine a man drew enough to fill a small keg, and then put into the cask the same quantity of water. Afterward he drew from the cask enough to fill the same keg, and then there were 41 gallons of pure wine in the cask. How much did the keg hold? Ans. 10 gallons. 20. There is a rectangular piece of land 75 rods long and 65 rods wide, and just within the boundaries there is a ditch of uniform breadth running entirely round the land. The land within the ditch contains 29 acres and 96 square rods. What is the width of the ditch? Ans. .5 of a rod. |