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90+ = 42 a
116 x 42 x = - 30 Changing signs
42 x2 116 x = 30
30 Dividing by 42
5 Reducing fractions de
21 To complete the second power of the first member, take one half of ji, which is i, and add its second power to both members.
841 5 841 x2
315 841 1156
21) Taking the root of both members,
21 The first value only will answer the conditions.
Ans. B sold 3 ells for a crown, and A 34. The learner may observe, that in raising 1 to the second power, I multiplied the numerator into itself, but expressed the power of the denominator by an exponent. This saved some work in this example. It may always be done when the number in the right hand member can be reduced to a fraction with the same denominator as the number added. In this case
could be reduced to 21ths. The was reduced thus
5 X 3 15 x 21 315
21) When the second member is a whole number, it can be reduced to a fraction with any denominator ; consequently this form may be used.
5. A man bought a certain number of sheep for 80 dollars; if he had bought 4 more for the same money, they would have come to him i dollar apiece cheaper. What was the number of sheep?
6. A merchant sold a quantity of brandy for £39 and gained as much per cent. as the brandy cost him. How much did it cost him?
Let x = the cost.
also 39 x = the gain.
7. Two persons, A and B, talking of their money, says A to B, if I had as many dollars as I have shillings, I should have as much money as you ; but if I had as many shillings as their number multiplied by itself, I should have three times as much money as you, and 63 shillings over. How much money had each ?
8. A colonel has a battalion of 1200 men, which he would draw up in a solid body of an oblong form, so that each rank may exceed each file by 59 men. What numbers must he place in rank and file?
9. A grazier bought as many sheep as cost him £60; out of which he reserved 15, and sold the remainder for £54, gaining 2 shillings a head by them. How many sheep did he buy, and what was the price of each ?
10. A person bought two pieces of cloth of different sorts; of which the finer cost 4s. a yard more than the other. For the finer he paid £18 ; but for the coarser, which exceeded the finer in length by 2 yards, he paid only £16. How many
yards were there in each piece, and what was the price of each?
11. A labourer dug two trenches, one of which was 16 yards longer than the other, for $77.60; and the digging of each cost as many dimes per yard, as there were yards in length. What was the length of each?
12. There are two square buildings, that are paved with stones each a foot square. The side of one building exceeds that of the other by 12 feet, and both their pavements taken together contain 2120 stones. What are the lengths of them separately.
13. A man bought two sorts of linen for $131. A yard of the finer cost as many shillings as there were yards of the finer. Also 30 yards of the coarser, (which was the whole quantity,) were at such a price, that 7 yards cost as much as a yard of the finer. How many yards were there of the finer, and what was the value of each piece ?
14. Two partners A and B gained £18 by trade. A's money was in trade 12 months, and he received for his principal and gain £26. Also B's money, which was £30, was in trade 16 months. What money did A put into trade ?
15. The plate of a looking glass is 18 inches by 12, and is to be framed with a frame, all parts of which are of equal width, and the area of the frame is to be equal to that of the glass. Required the width of the frame.
16. A and B set out from two towns, which were distant 247 miles, and travelled the direct road till they met. A went 9 miles a day; and the number of days, at the end of which they met, was greater by 3 than the number of miles which B went in a day. How many miles did each
17. A set out from C towards D, and travelled 7 miles per day. After he had gone 32 miles, B set out from D towards C, and went every day is of the whole journey; and after he had travelled as many days as he went miles in one day, he met A. What is the distance between the places C and D?
In this case both values will answer the conditions of the question.
18. A man had a field, the length of which exceeded the breadth by 5 rods. He gave 3 dollars a rod to have it fenced, which amounted to 1 dollar for every square rod in the field. What was the length and breadth, and what did he give for fencing it !
19. From two places at a distance of 320 miles, two persons, A and B, set out at the same time to meet each other. A tra. velled 8 miles a day more than B, and the number of days in which they met was equal to half the number of miles B went in a day. How many miles did each travel, and how far per day?
20. A man has a field 15 rods long and 12 rods wide, which he wishes to enlarge so that it may contain just twice as much; and that the length and breadth may be in the same proportion. How much must each be increased ?
In this example, the root can be obtained only by approximation.
21. A square court yard has a rectangular gravel walk round it. The side of the court wants 2 yards of being 6 times the breadth of the gravel walk; and the number of square yards in the walk exceeds the number of yards in the periphery of the court by 164. Required the area of the court ?
All equations of the second degree may be reduced to one of the following forms. 1.
+ px=9 2.
-PX = 9 3.
** + px=-9 4.
** -px=-9 After the equation has been brought to one of these forms, it may be solved by one of the following formulas, which are numbered to correspond to the equations from which they are derived.
The first equation and the first formula are sufficient for the whole, if pand q are supposed to be positive or negative quantities.
21. There are two numbers whose difference is 11%, and whose product is equal to 4 times the larger minus 9. What are the numbers ?
Let x = the larger;
— 113 x=4x — 9
2-32= -9. This equation is in the form of <* -px = -9, in which
and 52 10 4 100 x=1: + (* —97 = 1: + (***)*
= 7.8 +7.2. Or we may use the first formula, then
10' 4 100 x=18+ ( -97 = 11 € (449)
7.8 +7.2. Both values of x, being positive, will answer the conditions of the question.
Ans. By the first value the larger number is 15 and the smaller 38. By the second value of x, the larger is }, and the smaller -11.
Let the learner solve some of the preceding questions by the, formula.
XXXV. We shall now demonstrate that every equation of the second degree, necessarily admits of two values for the unknown quantity, and only two.