The nearest square root of 154 is 42 or 18, which is the root required within less than of one. 8. Required the square root of g. 9. What is the square root of 54? 10. Find the square root of . 11. Required the square root of 23. 12. What is the square root of a2 + x2? It has been shown, already, that no binomial is a perfect second power. The approximate root of a surd can be found by the common rule for extracting the square root of a compound quantity, thus: 13. Required the square root of 1 + x. 14. What is the square root of x2 z2? 15. Extract the square root of a1 + 1. + 256 alo xly 256 ale CHAPTER XI. EQUATIONS OF THE SECOND DEGREE SECTION I. Pure Equations. An equation of the second degree contains the second power of the unknown quantity. When the unknown quantity appears only in the second power, the equation is said to be pure. 1. What number is that, which, being multiplied by itself, and the product doubled, will give 162? Let the number. 2. A farmer, being asked how many cows he had, answered, that if the number were multiplied by 5 times itself, the product would be 720. How many had he? Let the number of cows. x2144, and x = 12. ANS. 12 cows. 3. A gentleman, being asked the price of his hat, answered, that if it were multiplied by itself, and 26 were subtracted from the product, the remainder multiplied by 5 would be 190. What was the price of the hat? Let the price of the hat. Then 5 2130 190, by the question. 130, or 320, by transposition. ANS. $8. 4. A gentleman, being asked the age of his son, replied, that if from the square of his age were subtracted his own age, which was 30 years, and the remainder were multiplied by his son's age, the product would be 6 times his age. How old was he? Let the son's age. Then, by the conditions of the question, (x2 30) x, or x3 30 x 6 x. 306, by dividing by x. x236, and x = 6. ANS. 6 years. 5. What two numbers are those, which are to each other as 3 to 4, and the difference of whose squares is 112? or 16 x — 9 x2 = 1792, by multiplication. ANS. 16 and 12. 6. There is a certain room, the sum of whose length and width is to its length as 5 to 3; and the same sum, multiplied by the length, is equal to the width multiplied by 60. What are the dimensions of the room? E. 5 x A. Then x + y =, by the question, 3 B. and 60 y = x2 + x y, that is, (x + y) x c. y = 2, by reducing equation A. a. = by comparing equations c and D. 3 3x2 = 120 x and 3 x 1202 x, by dividing by x. 3x+2x= 120, and x = 24. From these operations may be derived the following RULE for solving pure equations of the second degree: Find the value of the secona power of the unknown quantity, in the same manner as the value of the unknown quantity is found in simple equations; and then extract the square root of each member of the equation. Sometimes, as in the last question, the second power can be made to disappear by division. 7. A boy bought a number of oranges for 36 cents; and the price of an orange was to the number bought as 1 to 4. How many oranges did he buy, and what did he give apiece? 8. A merchant sold a quantity of flour for a certain sum, and at such a rate, that the price of a barrel was to the number of barrels as 4 to 5: if he had received 45 dollars more for the same quantity, the price of a barrel would have been to the number of barrels as 5 to 4. How many barrels did he sell, and at what price? 9. A gentleman exchanges a field, 81 rods long and 64 rods wide, for an equal quantity of land in the form of a square. What was the side of the square? 10. How long and wide is a rectangular field containing 864 rods, the width of which is equal to & of the length? 11. A certain street contains 144 rods of land; and if the length of the street be divided by its width, the quotient will be 16. How long and wide is the street? 12. A trader sold two pieces of broadcloth, which together measured 18 yards; and he received as many dollars a yard for each piece as it contained yards. Now, the sums received for the two were to each other as 25 to 16. How many yards were there in each piece? |