43. Prove that by Arts. (29–31), Alg. — { b2—(b2+02 —a2)"}, 2c (a+b+c)(b+c—a)(a+c—b)(a+b−c) 16 can be reduced to the form a+b+c 44. Prove that if, in the last example, s= the ex 2 pression may be reduced to the form s(s—a)(s—b)(s—c). 1 1 45. Show that++, except a=b. 46. Show that a3+17a2+a, except a=1. 47. If ax, show that (x+a)3—x37ax2. INVOLUTION AND EVOLUTION. 55. Required the square of x2-x+1. Ans. x4-2x+3x2-2x+1. 56. Required the cube of 1+2x+3x2. Ans. 1+6x+21x2+44x3 +63x1+54x3+27x6. 58. Extract the square root of 4x4-12x2+25x2-24x +16. Ans. 2x2-3x+4. 61. Extract the cube root of x-6x+15x1—20x3+15x2 -6x+1. Ans. x2-2x+1. 62. If two numbers differ by 1, prove that the difference of their squares is the sum of the two numbers. 63. If two numbers differ by any number (a), prove that the difference of their squares is the sum of the numbers multiplied by their difference (a). 64. If the sum of two fractions=1, prove that their difference is equal to the difference of their squares. 65. Prove that the square of any odd number diminished by 1 is divisible by 8. 66. Prove that the product of two odd numbers is odd; and that the product of two even numbers, or of an even and an odd number is even. EXERCISES IN SIMPLE EQUATIONS. 67. Given 8x-4-13-x, to find x. 3x Ans. x=2. 68. Given 2x+7+=6x-23, to find x. Ans. x=12. PROBLEMS PRODUCING SIMPLE EQUATIONS. 87. How much money have I in my pocket, when the fourth and fifth of the same together amount to 11s. 3d. Ans. L.1, 5s. 88. In a company of 266 persons, consisting of officers, merchants, and students, there were four times as many merchants, and twice as many officers, as students. How many were of each class? Ans. 38 students, 152 merchants, and 76 officers. 89. Divide the number a into three such parts, that the second may be m times, and the third n times as great as the first. Ans. a ma na 1+m+n' 1+m+n' l+m+n' 90. A father gives to his five sons L.1000, which is to be divided among them according to their ages, so that each of the four elder may receive L.20 more than his next younger brother. How much will the youngest receive? Ans. L.160. 91. A courier who goes a miles a-day, started n days before another, who goes b miles a-day. In what time will the second overtake the first? b-a na Ans. days. 92. A person paid the sum of L.2, 13s. in crowns and shillings, using 17 pieces in all. How many of each sort was used? Ans. 9 crowns and 8 shillings. 93. One of my acquaintances is now 40 years old, and his son 9 years. In how many years will this man, who is now more than four times as old as his son, be only twice as old? Ans. In 22 years. 94. A transcriber was asked how many sheets he wrote weekly: he answered, "I only work four hours a-day, and cannot finish 70 sheets, which I could wish to do; but if I could work ten hours a-day, then I should write weekly exactly as many above 70 sheets as I now write less than that number." How many sheets did he write weekly? Ans. 40. 95. Find two numbers, such that if the first be multiplied by 2, and the second by 5, the sum of the products will be 31; and if the first be multiplied by 7, and the other by 4, the sum of the products will be 68. Ans. First 8, and second 3. 96. It is required to find a fraction, such that if 3 be subtracted from the numerator and denominator, the value of the remaining fraction will be, and if 5 be added to M each of its terms, its value will be. What is the fraction? Ans. 97. The sum of two numbers is a, the difference of b. What are the numbers? their squares 99. A, B, and C, owe L.2190 amongst them, and no one of them can pay this sum alone. But when they unite, it can be done in the following ways: by B putting of his property to all A's; or by C putting of his property to that of B; or by A putting of his property to that of C. How much did each possess? Ans. A L.1530, B L.1540, C L.1170. 100. Required to find three numbers, which possess the following properties:-If 6 be added to the first and second, the sums are to one another as 2 to 3; if 5 be added to the first and third, then the sums are to one another as 7 to 11; but if 36 be subtracted from the second and third, the remainders are to one another as 6 to 7. What are the numbers? Ans. 30, 48, and 50. 26 109. If (a—b)x2—(a+b)x+2b=0. x=1, or 12 110. If x+4+(+4)* = = =±5, or ±4√2. 111. If x7x3-8. X- -4 x=2, or —1. |