tor is not a complete power, the root whose second power is nearest to 5 must be taken, which is 2. The difference between and the true square root of, is less than of a unit. When both the numerator and the denominator of a fraction are multiplied by the same quantity, its value is not altered; and if both terms of this fraction be multiplied by any perfect power, a nearer approximation to the true root will be obtained. Let them be The square root of 81 is 9, and the nearest square root of 45 is 7; so that expresses the value of within part of a unit. Again, let be multiplied by 144, which is the second power of 12. In general, the larger the power by which the terms of a fraction are multiplied, the nearer to the true root will be the approximation. 7. What is the square root of 34? Reduce this mixed number to an improper fraction, and proceed as before. The nearest square root of 4 is 42 or 14, which Is the root required within less than of one 8. Required the square root of g. 9. What is the square root of 5 10. Find the square root of 3. 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 14. What is the square root of x2 15. Extract the square root of aa + 1. 16. What is the square root of 17. Extract the square root of 7641. + x. 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. x2 = 144, 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? 4. A gentleman, being asked the age of his son replied, that if from the square of his age were sub tracted 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. x2-306, by dividing by x. ANS. 6 years. 5. What two numbers are those, which are to each other as 3 to 4, and the difference of whose squares 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? Let x = the length, and y = the width of the room. x ▲. Then x + y = 5*, by the question, A. 3 B. and 60 y = x2 + xy, that is, (x + y) x and 3 x 2x 27, by reducing equation a. x2 by reducing equation B. by comparing equations c and D. 2 x2, 120 - 2 x, by dividing by x. 3x + 2 x = 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; |