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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 go ?

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 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 travelled 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 approxi

mation.

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.

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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.

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The first equation and the first formula are sufficient for the whole, if p and q are supposed to be positive or negative quantities.

22. There are two numbers whose difference is 113, and whose product is equal to 4 times the larger minus 9. What are the numbers?

Let the larger;

then x-11= the smaller.

x2-11x=4x-9

x2-7829.

This equation is in the form of x2-px-q, in which

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—9)*

=

100

and q = 9.

x = 78 ± (*!!!! — 9)* = 78 ± (!!!)* = 7.8 ±7.2.

Or we may use the first formula, then

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184

x = 78 ± (%%%' — 9) * = 78 ± ('?!?!) *

6084
100

= 7.872.

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 3. By the second value of x, the larger is, and the smaller 11.

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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.

Let us take the general equation.

x2+px=q.

This, we have seen, may represent any equation whatever of the second degree, p and q being any known quantities and either positive or negative. If p = 0 the equation becomes x2 = 9,

which is a pure equation or an equation with two terms.

If we make the first member of the equation x2+px=q, a complete second power, by the above rules, it becomes

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The first member of this equation is the difference of two second powers, which, Art. XIII, is the same as the product of the sum and difference of the numbers.

The sum is x ++m, and the difference is x + 1⁄2

and their product is

(x + 2 − m) (x+2+m) = 0.

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In this equation, the first member consists of two factors, and the second is zero. Now the first member of the above equation will be equal to zero, if either of its factors is equal

to zero. For if any number be multiplied by zero, the product is zero.

Making the first factor equal to zero,

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Either of these values of x must answer the conditions of the equation.

N. B. Though either value answers the conditions separately, they cannot be introduced together, for being different, their product cannot be x2.

Instead of m put its value, and the values of x become

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which are the values we had obtained above. (This demon stration is essentially that of M. Bourdon.)

Discussion.

Let us take again the general equation.

− 2 = ( 9 + 2 2 ) + .

Since the expression contains a radical quantity, that is, a quantity of which the root is to be found, in order to be able to find the value of it, we must be able to find the root either exactly or by approximation. Now there is one case in which

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