PROBLEMS PRODUCING QUADRATIC EQUATIONS. Prob. 1. A merchant has a piece of cotton cloth, and a piece of silk. The number of yards in both is 110; and if the square of the number of yards of silk be subtracted from 80 times the number of yards of cotton, the difference will be 400. How many yards are there in each piece? Let the yards of silk. Then 110-x-the yards of cotton. By supposition, That is, 400-80×(110-x)-x 400-8800-80x-x2 Transp. & unit. terms, a2 +80x=8400 Compl❜g the square, x2 +80x+1600=1600+8400=10000 Extracting and transp. x=-40±√10000=-40±100 The first value of x, is -40+100-60, the yards of silk; And 110-x=110-60-50, the yards of cotton. The second value of x, is -40-100=-140; but as this is a negative quantity, it is not applicable to goods which a man has in his possession. Prob. 2. The ages of two brothers are such, that their sum is 45 years, and their product 500. What is the age of each? Let x=one of the ages. Then 45-x=the other. Changing all the signs, x2-45x=-500 xx (45-x)=500 45x-x2 500 One of the ages then is 25 years, and the other 20. Prob. 3. To find two numbers such, that their difference shall be 4, and their product 117. Let x=one number, and x+4=the other. (x+4)×x=117 By the conditions, This reduced, gives, x=-2± √121=-2±11. One of the numbers therefore is 9, and the other 13. Prob. 4. A merchant having sold a piece of cloth which cost him 30 dollars, found that if the price for which he sold it were multiplied by his gain, the product would be equal to the cube of his gain. What was his gain? Let x=the gain. Then 30+x=the price for which the cloth was sold. As the last answer is negative, it is to be rejected as inconsistent with the nature of the problem, (Art. 320.) for gain must be considered positive. Prob. 5. To find two numbers, whose difference shall be 3, and the difference of their cubes 117. Let x=the least number. Expanding (x+3)3 (Art.217.) 9x2+27x=117-27=90 ́ By supposition, (x+3)3-x3=117 Dividing by 9, x2+3x=10 Completing the square, x2 +3x+12=2+10=49 The two numbers, therefore, are 2 and 5. Prob. 6. To find two numbers, whose difference shall be 12, and the sum of their squares 1424. Ans. The numbers are 20 and 32. Prob. 7. Two persons draw prizes in a lottery, the difference of which is 120 dollars, and the greater is to the less, as the less to 10. What are the prizes? Prob. 8. What two numbers are those whose sum is 6, and the sum of their cubes 72? Ans. 2 and 4. SUBSTITUTION. 321. In the reduction of Quadratic Equations, as well as in other parts of algebra, a complicated process may be rendered much more simple, by introducing a new letter which shall be made to represent several others. This is termed substitution. A letter may be put for a compound quantity as well as for a single number. Thus in the equation x2 -2ax=2+√86–64+h, we may substitute b, for +86-64+h. The equation will then become x2-2ax=b, and when reduced will be x=a± √a2 +b. After the operation is completed, the compound quantity for which a single letter has been substituted, may be restored. The last equation, by restoring the value of b, will become x=α± √a2 + +√86-04+h. Reduce the equation ax-2x-d=bx-x2 Transp. and uniting terms, x2+ax-bx-x=d By art. 120, Substituting h for (a−b−1),x2 +hx=d Completing the square, h2 h2 x2+hx + 4 = 4+d SECTION XI. SOLUTION OF PROBLEMS WHICH CONTAIN TWO OR MORE UNKNOWN QUANTITIES. DEMONSTRATION OF THEOREMS. ART. 322. IN the examples which have been given of the resolution of equations, in the preceding sections, each problem has contained only one unknown quantity. Or if, in some instances, there have been two, they have been so related to each other, that both have been expressed by means of the same letter. (Art. 195.) But cases frequently occur in which several unknown qaantities are introduced into the same calculation. And if the problem is of such a nature, as to admit of a determinate answer, there will arise from the conditions, as many equations independent of each other, as there are unknown quantities, Equations are said to be independent, when they express different conditions; and dependent, when they express the same conditions under different forms. The former are not convertible into each other. But the latter may be changed from one form to the other, by the methods of reduction which have been considered. Thus b-x=y, and b=y+x, are dependent equations, because one is formed from the other by merely transposing x. 323. In solving a problem, it is necessary first to find the value of one of the unknown quantities, and then of the others in succession. To do this, we must derive from the equations which are given, a new equation, from which all the unknown quantities except one shall be excluded. Suppose the following equations are given. 1. x+y=14 2. x-y=2. If y be transposed in each, they will become 1. x=14-y 2. x=2+y. |