equation 2x2+3x=65, the sign of the second term 3x only, is changed, because (-x)2=x2. 3 23 Therefore, instead of obtaining =→ ㄓ we would find 4 -5, values which only differ from the preceding by their signs. Hence, we may say that the nega. tive solution 13 considered independently of its sign, satisfies this new enunciation, viz.: To find a number such, that twice its. square, diminished by three times this number, shall give 65. In fact, we have 240 cents. 2. A certain person purchased a number of yards of cloth for If he had received 3 yards less of the same cloth, for the same sum, it would have cost him 4 cents more per yard. How many yards did he purchase? Let x= the number of yards purchased. If, for 240 cents, he had received 3 yards less, that is x-3 yards, the price per yard, in this hypothesis, would have been repre But, by the enunciation this last cost would ex The value x=15 satisfies the enunciation; for, 15 yards for 240 240 cents, gives or 16 cents for the price of one yard, and 12 yards for 240 cents, gives 20 cents for the price of one yard, which exceeds 16 by 4. As to the second solution, we can form a new enunciation, with which it will agree. For, go back to the equation, and change r intox, it becomes, an equation which may be considered the algebraic translation of this problem, viz.: A certain person purchased a number of yards of cloth for 240 cents: if he had paid the same sum for 3 yards more, it would have cost him 4 cents less per yard. How many yards did he purchase? Ans. x=12, and x=- -15. REMARK. Hence the principles of (Arts. 104 and 105.) are confirmed for two problems of the second degree, as they were for all problems of the first degree. 3. A merchant discounted two notes, one of $8776, payable in nine months, the other of $7488, payable in eight months. He paid $1200 more for the first than the second. At what rate of interest did he discount them? To simplify the operation, denote the interest of $100 for one month by x, or the annual interest by 12x; 9a and 8x are the interests for 9 and 8 months. Hence 100+9x, and 100+8x, represent what the capital of $100 will be at the end of 9 and 8 months. Therefore, to determine the present values of the notes for $8776, and $7488, make the two proportions, and the fourth terms of these proportions will express what the mer chant paid for each note. Hence, we have the equation or, observing that the two members are divisible by 400, Clearing the fraction, and reducing, it becomes, Reducing the two terms under the radical to the same denomi To obtain the value of 12x to within 0,01, we have only to extract the square root of 5306404 to within 0,1, since it is afterwards to be divided by 18. The positive value, 12x=5,86, therefore represents the rate of interest sought. As to the negative solution, it can only be regarded as connected with the first by an equation of the second degree. By going back to the equation, and changing x into -x, we could with some trouble, translate the new equation into an enunciation analogous to that of the proposed problem. 4. A man bought a horse, which he sold after some time for 24 dollars. At this sale, he loses as much per cent. upon the price of his purchase, as the horse cost him. What did he pay for the horse? Let x denote the number of dollars that he paid for the horse, x-24 will express the loss he sustained. lost x per cent. by the sale, he must have lost x But as he 100 upon each dollar, and upon x dollars he loses a sum denoted by Both of these values satisfy the question. For, in the first place, suppose the man gave $60 for the horse and sold him for 24, he loses 36. Again, from the enunciation, he which reduces to 36; therefore 60 satisfies the enunciation. If he paid $40 for the horse, he loses 16 by the sale; for, he should lose 40 per cent. of 40, or 40 X therefore 40 verifies the enunciation. 40 which reduces to 16; 100' 5. A grazier bought as many sheep as cost him £60, and after reserving fifteen out of the number, he sold the remainder for £54, and gained 2s a head on those he sold : how many did he buy? Ans. 75. 6. A merchant bought cloth for which he paid £33 15s, which he sold again at £2 8s per piece, and gained by the bargain as much as one piece cost him: how many pieces did he buy? Ans. 15. 7. What number is that, which, being divided by the product of its digits, the quotient is 3; and if 18 be added to it, the digits will be inverted? Ans. 24. 8. To find a number such that if you subtract it from 10, and multiply the remainder by the number itself, the product shall be 21. Ans. 7 or 3. 9. Two persons, A and B, departed from different places at the same time, and travelled towards each other. On meeting, it appeared that A had travelled 18 miles more than B; and that A could have gone B's journey in 153 days, but B would have been 28 days in performing A's journey. How far did each travel? Discussion of the General Equation of the Second Degree. 141. As yet we have only resolved problems of the second degree, in which the known quantities were expressed by particular numbers. To be able to resolve general problems, and interpret all of the results obtained, by attributing particular values to the given quantities, it is necessary to resume the general equation of the second degree, and to examine the circumstances which result from every possible, hypothesis made upon its co-efficients. This is the object of the discussion of the equation of the second degree. 142. A root of an equation of the second degree, is such a number as being substituted for the unknown quantity, will satisfy the equation. |
