Ans. x-9, or -5. 1. Given x2-4x+45; to find x. Ans. x 2, or -23. 3. Given x2-50-5-6x ; to find x. x; 14-x Ans. x-5, or -11. 4. Given 4x-14= ; to find x. Ans. x=4, or -7. x+1 7. Given 4x2+; to find x. Ans. x1, or -8. 8. Given 2x2+2x+6=304; to find x. 22 Ans. x 3, or -4. Ans. 11, or -10. 10. Given=24x-700; to find x. Ans. x=70, or 50. 5 QUESTIONS PRODUCING QUADRATIC EQUATIONS. 1. What two numbers are those whose difference is 15, and half of whose product is equal to the cube of the less? Ans. 3 and 18. 2. What two numbers are those whose sum is 100, and whose product is 2059? Ans. 71 and 29. 3. Find two numbers, so that their difference may be 8, and their product 240. Ans. 20 and 12. 4. Having sold a piece of cloth for L.24, I gained as much per cent. as the cloth cost me; what was its prime cost? Ans. L.20. 5. A grazier bought as many oxen as cost him L.480, and retaining 6 to himself, sold the remainder for the same sum, by which he gained L.4 a head on those sold. How many oxen did he buy, and what did he pay for each? Ans. 30 oxen, at L.16 each. 6. A labourer dug two trenches, one of which was 4 yards longer than the other, for L.20, and each trench cost as many shillings a-yard as there were yards in its length. How many yards were in each? Ans. 12 yards and 16 yards. 7. The plate of a looking-glass is 24 inches by 16; it is to be framed by a frame of uniform width throughout, whose surface shall be equal to the surface of the glass. Required the breadth of the frame. Ans. 4 inches. 8. There are three numbers in geometrical progression. The sum of the first and second is 10, and the difference of the second and third is 24. What are the numbers? Ans. 2, 8, 32. 9. A and B set off at the same time to a place at the distance of 300 miles. A travels at the rate of one mile an hour faster than B, and arrives at his journey's end 10 hours before him. At what rate did each travel per hour? Ans. A travelled 6 miles per hour, and B travelled 5. 10. A and B distribute L.1200 each among a certain number of persons. A relieves 40 persons more than B, and B gives L.5 a-piece to each more than A. How many persons were relieved by A and B respectively? Ans. 120 by A, and 80 by B. 11. A person bought cloth for L.33, 15s., which he sold again at L.2, 8s. per piece, and gained as much by the bargain as one piece cost him. Required the number of pieces. Ans. 15. 12. A company dine together at an inn for L.3, 15s. One of them was not allowed to pay, and the share of each of the rest was, in consequence, half-a-crown more than if all had paid. How many were in the company. Ans. 6. 13. A draper bought two pieces of cloth for L.3, 8s. The one was 6 yards longer than the other, and each of them cost as many shillings a-yard as there were yards in the piece. What was the length of each? Ans. 2 yards and 8 yards. 14. Two girls carry 100 eggs to market. One of them had more than the other, but the sum which each received was the same. The first says to the second, if I had had as many eggs as you, I should have received 15 pence. The other answers, if I had had your number, I should have received 63 pence. How many eggs had each, and what did each receive? Ans. The first girl had 40, and the second 60, and each received 10 pence. 15. Find three numbers having equal differences, so that their sum may be 9, and the sum of their fourth powers 707. Ans. 1, 3, and 5. QUADRATIC EQUATIONS, WITH TWO UNKNOWN QUANTITIES. 83. In solving quadratic equations with two unknown quantities, it is necessary, frequently, to find a value of one f the unknown quantities in terms of the other unknown, nd then substitute this value, instead of it, in the other quation, and then solve this equation for the other unnown quantity; then the remaining unknown quantity easily be found, either by substituting this value, or therwise, as may be found most convenient. If the sum, r the sum of the squares of the two quantities be given, nd their product be either given or can be found, the sum nd difference of the quantities can be found, and then eir values determined by the solution of a simple equa on. EXAMPLE 1. Given x+y=21, and nd y. From the second equation we find x-2y2; substituting his value instead of x in the first it becomes 2y2+y=21. Hence 16y2+8y+1=169, by comp. sq. 4y+1=13, extract the root. These values substituted instead of y in the first equa ion give x=18, or 241. EXAMPLE 2. Given x+y=10, and xy=24. x2+2xy+y2=100. 4xy=96. x2-2xy + y2=4. .. 2x=12 (a)+(b). .. 2y=8 (b)-(a) and y=4. EXAMPLE 3. There are two numbers, the difference of whose cubes is 117, and the difference of the numbers themselves is 3, what are those numbers? Let x=the greater, and y=the less; then by the ques Divide the first equation by the second, and there results x2+xy+y2=39 (a). Square of the 2d x2-2xy+y2=9 (b). .. 3xy-30, by subtraction. x2+2xy+y2=49 (a)+(c). x+y=7, by extract. root. But x-y=3. Hence x=5, and y=2. EXAMPLE 4. What two numbers are those whose sum multiplied by the greater is 77, and whose difference multiplied by the less is 12? Let x the greater, and y=the less. And (x-y)y=xy—y2=12. Assume xvy, then by substituting this value instead of x in each of the equations, they become v2y2+vy2=77. And vy-y2-12. The first of these being divided by the second, gives v2y2+vy2 _v2+v = 77 vy2-y3 V Hence 12 12v2+12v=77v-77. Either of these values will fulfil the conditions of the question. The first gives x=√2, and y=√2, and the second gives x=7, and y=4. EXERCISES. 1. Given (x+y=12 to find x and y. 1 xy=32J 2. Given x-y=7 to find x and y. | xy=18] 3. Given (x2+y=169) to find x and y. x-y=7 Ans. x=8, y=4. Ans. x=9, y=3. x2-y2=721 to find x and y. x2+3y2=108) x+2y=151 to find x and y. x2+4y2=113 Ans. a=8 or 7, y=3 or 4. x+4y=63 to find ≈ and 4. Given 5. Given 6. Given 4x 5 xy=96 to find x and y. y. Ans. x=15, y=12. Ans. x 8, y=12. Ans. x=10, y=8. x2-xy=201 to find x and y. 2 xy-y2=16) 9. The sum of two numbers multiplied by the greater is 240, and their sum multiplied by the less is 160; what are these numbers? Ans. 12 and 8. 10. What two numbers are those, of which the sum multiplied by the greater is equal to 220, and the difference multiplied by the less is equal to 18? Ans. 11 and 9, or 10/2 and √2. 11. Find three numbers such that their sum multiplied by the first may be 48, their sum multiplied by the second may be 96, and their sum multiplied by the third may be Ans. 3, 6, 7. 112. 12. Find two numbers such that five times their difference may be equal to four times the less, and the square of the greater, together with four times the square of the less, may be 181. Ans. 9 and 5. 13. What two numbers are those, the sum of whose squares is 34, and the sum of whose fourth powers is 706? Ans. 5 and 3. 14. What fraction is that which is double of its square, and whose numerator increased by one, and the sum multiplied by its numerator, is equal to the denominator diminished by one, and the remainder multiplied by the denominator? Ans.. SURDS. 84. Those roots which cannot be expressed in finite terms, are called surds or irrational quantities. 3 n m m Thus the √/2, 3/a2, ~/aTM or 23, a3, x, are surds, for their values cannot be expressed in finite terms. All surds may be expressed by means of fractional exponents, in which it is evident that the value of the surd will not be altered by multiplying or dividing both numerator and denominator of the fractional exponent by the same number; thus 83–8¤—(83)*, mr = m mr m or a"ar for a raised to mr ; the rth power, is απ and the rth root of this is a r but a quantity raised to the rth power, and then the rth root extracted, is the quantity itself; hence a surd is not altered in value by multiplying both numerator and denominator of its exponent by the same number, and since a has al m mr ready been shown to be equal to a", the surd is not altered in value by dividing both numerator and denominator of its |