Ꮖ by x ..... 101 7x+2x+2=5x2+5x. By transposition, 5 x2-4x=2. By RULE II. 100x3-80x+16=40+16=56. and 10x=/56+4=7.48+4=11.48 11.48 ..x= =1.148. 10 Let 2bx-cx=d. By RULE II. multiply 16 b2x2 - 8 b c x+c=8bd+c2. by 8b, and add c, S square root } 4 bx c = √8bd+c .. 4bx=8bd+c2+c. XXII. On the Solution of Questions producing Quadratic Equations. 84. In the solution of Questions which involve quadratic equations, sometimes both, and sometimes only one of the values of the unknown quantity, will answer the conditions required. This is a circumstance which may always be very readily determined by the nature of the question itself. QUESTION 1. To divide the number 56 into two such parts, that their product shall be 640. Let x=one part, then 56-x=the other part, and x(56-x)=product of the two parts. Hence, by the question, x(56-x)=640, or 56x-x640. By transposition, x-56x=-640. By completing the square, (RULE I.) x2-56x-784-784-640=144; ..x-28=√144±12, and x 2812=40 or 16. In this case it appears that the two values of the unknown quantity are the two parts into which the given number was required to be divided. QUESTION 2. There are two numbers whose difference is 7, and half their product plus 30 is equal to the square of the lesser number. What are the numbers? Let x=the lesser number, then x+7=the greater number, .. 2x−7=√/289=17 2x=17+7=24, or x=12 lesser number; hence x+7=12+7=19 greater number. QUESTION 3. To divide the number 30 into two such parts, that their product may be equal to eight times their difference. and 30-x-x or 30-2x=their difference. Hence, by the question, x(30-x)=8(30—2x), or 30x-x=240-16x. By transposition, x2-46x=-240. Complete the square, (RULE I.) x2-46x+529-529-240=289; ..x-23=±√289±17, = and x2317=40 or 6 lesser part; 30-x=30- 6=24=greater part. In this case, the solution of the equation gives 40 and 6 for the lesser part. Now as 40 cannot possibly be a part of 30, we take 6 for the lesser part, which gives 24 for the greater part; and the two numbers, 24 and 6, answer the conditions required. QUESTION 4. A person bought cloth for 337. 15s. which he sold again at 21. 8s. per piece, and gained by the bargain as much as one piece cost him; Required the number of pieces. Let Let x the number of pieces. 675 Then = number of shillings each piece cost, and 48x=number of shillings he sold the whole for; .. 48x-675=what he gained by the bargain. Hence, by the question, 48x-675= and division, Complete the 225 50625 675 By transposition + QUESTION 5. A and B set off at the same time to a place at the distance of 150 miles. A travels 3 miles an hour faster than B, and arrives at his journey's end 8 hours and 20 minutes before him. At what rate did each person travel per hour? Let x-rate per hour at which B travels. But A is 8 hours 20 minutes (8 hours) sooner at his journey's end than B; |