A's money (expressed in dollars, and considered merely as a number) by B's; and B's money by C's, and add both products to the united fortunes of all three, we shall have 8832. How much had each?

Ans. A $40, B $72, C $80.

8. A person buys some pieces of cloth, at equal prices, for $60. Had he received three more pieces for the same sum, each piece would have cost him $1 less. How many pieces did he buy? Ans. 12 pieces.

9. Two travellers, A and B, set out at the same time, from two different places, C and D; A, from C to D; and B, from D to C. On the way they met, and it then appears that A had already gone 30 miles more than B, and, according to the rate at which they travel, A calculates that he can reach the place D in 4 days, and that B can arrive at the place C in 9 days. What is the distance between C and D ? Ans. 150 miles.

10. Divide the number 60 into two such parts, that their product may be to the sum of their squares, in the ratio of 2 to 5. Ans. 20 and 40.

11. A grazier bought as inany sheep as cost him $150, and after reserving 15 out of the number, he sold the remainder for $135, and gained $¦ a head. How many. sheep did he buy? Ans. 75 sheep.

12. What number is that, which, when divided by the product of its two digits, the quotient is 3; and if 18 be added to it, the digits will be inverted?

Ans. 24.

13. Two partners, A and B, gained $140 by trade; A's money was 3 months in trade, and his gain was $60 less than his stock ; and B's money which was $50 more than A's, was in trade 5 months; what was A's stock? Ans. $100.

14. It is required to divide the number 24 into two such parts, that their product may be equal to 35 times their difference. What are the parts?

Ans. 10 and 14.

15. A company at a tavern had $60 to pay for their reckoning; but, before the bill was settled, two of them left the room, and then those who remained had $1 a-piece more to pay than before. How many were there in the

company?

16. There are two numbers whose and half their product is equal to the number. What are those numbers ?

Ans. 12.

difference is 15, cube of the less

Ans. 3 and 18.

17. A merchant bought a certain number of pieces of cloth, for $200, which he sold again at $10 per piece, and gained by the bargain as much as one piece cost him. What was the number of pieces?

Ans. 20 pieces.

18. A and B together, agree to dig 100 rods of ditch. for $100. That part of the ditch on which A was employed was more difficult of excavation than the part on which B was employed. It was therefore agreed that A should receive for each rod 25 cents more than B received for each rod which he dug. How many rods

must each dig, and at what prices, so that each may re

PROPERTIES OF THE ROOTS OF QUADRATIC EQUATIONS.

(122.) We have seen that all quadratic equations can be reduced to this general form.

This, when solved by the Rule under Art. 118, gives

real values of a, it follows that the sign of the expres

a2

sion+b, depends upon the value of b.

4

(124.) When b is positive, or when b is negative and

a2

a2

less than then will+b be positive, and consequen

4'

4

+b will be real.

(125.) When b is negative and numerically greater

a2

+b, then will both values of x be real and nega

4

tive.

α

2. When a is either positive or negative, and

is nu

2

a2

merically less than

-+b, then will both values of

4

x be real, the one positive, and the other negative.

3. When a is negative and

α

a2

than

4

+b, then both values of x will be real and

α

2

is numerically greater

In this case both values of x are imaginary for all values of a.

(126.) When b is negative and numerically equal to

then both values of x become

α

2

(127.) If we add together the two values of x, we

That the sum of the roots of the quadratic equation x2+ax-b is equal to -a.

And the product of the roots is equal to -b.

(128.) We have seen that every quadratic equation, when solved, gives two values for the unknown quantity. These values will both satisfy the algebraic conditions, and sometimes they will both satisfy the particular conditions of the problem, but in most cases but one value of the unknown is applicable to the problem; and the value to be used must be determined from the nature of the question.

We will illustrate this principle by the solution of some particular questions.

1. Find a number such that its square being subtracted from five times the number, shall give 6 for remainder.