m Ex. 9. Given xm + bx2 = a, to find the value of î. Required the positive values of x in the following equations : On the application of Quadratic Equations to the Solution of Problems. 94. Let the conditions of the given problem be first translated into the language of Algebra, as in Simple Equations; there will thence arise an equation from which the value of the unknown quantity may be easily determined, by the rule stated in No. 92. In some instances, both values of the unknown quantity will answer the conditions of the proposed problem, in others only one, as will readily appear from the nature of the problem itself. Ex. 1. It is required to divide the number 18 into two such parts that their product may be 56. Let x Then 18 the one part. - the other. And (18-x)x = 18x-x2= product of the two parts. Hence 18x256, (by Quest.;) or changing the signs, x2 18x= -56. Comp. square, x2-18x+8156 +81=25. 56+81 √255. Transp. x5+ 9 14 or 4, where it appears that the two values of x are just the two parts into which the given number was required to be divided. Ex. 2. The difference of two numbers is 5, and of their product is equal to four times the less number plus 20. Required the numbers, Ex. 3. The product of two numbers is 60, and if 5 be subtracted from the greater, and 2 added to the less, the product of the sum and remainder will also be 60. Required the numbers. And, by cancelling the 60x, since it appears on both sides with the same sign, we have, Ex. 4. A mirror, 24 inches by 18, is to be so framed, that the area of the front of the frame may be of that of the mirror. Required its breadth. Then 24+2x= length of the frame. area of one of the side pieces. Now, since there are two of them, (24x + 2x2)2= 48x+4x2 area of the two side pieces, 18x area of one of the end pieces, and 18x × 2, 36x = area of the two end pieces. Hence, 48x+4x2 + 36x = 4x2 + 84x= area of the whole frame. Now, 4x2 +84x — 2 of 432, (the area of the mirror.) == 3 27 Hence, x = 2 21 6 Ex. 5. A and B set distance of 300 miles. = = 3 inches, the breadth of the frame. 2 2 off at the same time to a place at the A travels at the rate of one mile an hour faster than B, and arrives at his journey's end ten hours before him. What were their respective rates of travelling? Let x = A's rate of travelling per hour. B's = = time taken on the journey by A. B. EXERCISES. Ex. 1. What two numbers are those whose sum is 17, and whose difference, multiplied by the less, is equal to 30? Ans. 11 and 6. Ex. 2. A and B distribute each 20s. among a certain number of beggars. A relieves 20 persons more than B; but B gives 2d. to each more than A does. How many persons were relieved by A and B respectively? Ans. 60 by A, and 40 by B. Ex. 3. Two persons, A and B, were despatched at the same time to a place at the distance of 90 miles; the former of whom travelled one mile an hour faster than the other, and arrived at the journey's end one hour sooner. At what rate did each person travel? Ans. A 10, and B 9 miles per hour. Ex. 4. What number is that whose square is greater than its simple power by 110? Ans. 11, Ex. 5. A gentleman ordered 10 guineas to be distributed among a certain number of poor people; but, before the distribution took place, there came in six claimants more, by which means, each of the former received 4s. less than he or she would otherwise have done. How many were there at first? Ans. 15. Ex. 6. There is a certain number, consisting of two places of figures, which, when divided by the sum of its digits, gives a quotient greater by 2, than the left hand digit; but if the digits be inverted, and then divided by a number greater by unity than the sum of the digits, the quotient will be greater by 2 than the preceding quotient. Required the number. Ans. 24. Ex. 7. A gentleman has a garden in the form of an oblong, whose length is 20 yards and breadth 15. He wishes to have a walk made round it, whose area shall be of that of the whole garden. What must be the breadth of the walk? Ans. 3.44 Feet. He Ex. 8. A person bought a horse for a certain sum. afterwards sold it for L. 144, and gained exactly as much per cent. as the horse cost him. Required the cost. Ans. L. 80. |