number of rods round it. Required the price of the farm, and its length and breadth.

Ans. Price, $1600; length 160 rods, breadth 40 rods.

10. I have two cubic blocks of marble, whose united lengths are 20 inches, and contents 2240 cubic inches. Required the surface of each.

Ans. 864 square inches; 384 square inches.

11. A's and B's shares in a speculation altogether amount to $500; they sell out at par, A at the end of 2 years, B of 8, and each receives in capital and profits $297. How much did each embark?

Ans. A, $275; B, $225.

12. A farmer bought as many sheep as cost him $300; after reserving 15 out of the number, he sold the remainder for $270, and gained $0.50 per head on them. How many sheep did he buy? Ans. 75.

13. A person has $1300, which he divides into two portions, and loans at different rates of interest, so that the two portions produce equal returns. If the first portion had been loaned at the second rate of interest, it would have produced $36, and if the second portion had been loaned at the first rate of interest, it would have produced $49. Required the rates of interest.

Ans. 7 and 6 per cent.

14. Two men, A and B, bought a farm of 200 acres, for which they paid $200 each. On dividing the land, A says to B, "If you will let me have my part in the sitnation which I shall choose, you shall have so much more and than I that mine shall cost 75 cents per acre more than yours." B accepted the proposal. How much land did each have, and what was the price of each per acre?

Ans. A, 81.867 acres, at $2.443; B, 118.133 acres, at $1.693.

15. A and B start at the same time from two distant towns. At the end of 7 days, A is nearer to the half-way house than B is, by 5 miles more than A's day's journey. At the end of 10 days they have passed the half-way house, and are distant from each other 100 miles. Now it will take B three days longer to perform the whole journey than it will A. Required the distance of the towns, and the rate of walking of A and B.

Ans. Distance of the towns, 450 miles; A's rate, 30 miles, and B's rate, 25 miles, per day.

THEORY OF QUADRATIC EQUATIONS.

292. Every complete quadratic equation, it has been shown (Art. 274), may be reduced to the form

x2 + px = 9,

where p and q represent known quantities, positive or negative, integral or fractional. This form may also represent a pure quadratic, by considering p = 0.

The values of x in this equation are

and, since no other values necessarily result from the general equation, we infer that

Every equation of the second degree has TWO ROOTS, and only

two.

293. If we add the two roots of the general equation (Art. 292), their sum is p, and if we multiply them together, their product is

1. The algebraic SUM of the two roots of a quadratic equation is equal to the coefficient of the second term, with its sign changed.

2. The PRODUCT of the two roots is equal to the second memher, with its sign changed.

294. Representing by r and the two roots of the quadratic equation

we have

that is,

x— r = 0, or xrl = 0.

Multiplying the last two expressions together, we have

(x − r) (x — r') = 0,

x2 - (r + r) x + rr = 0.

But, by Article 293,

whence,

That is,

r + r = -p, and rr =

x2 + px − q = (x − r) (x — r') = 0.

If all the terms of a quadratic equation be transposed to the first member, it may be resolved into the two binomial factors formed by subtracting each of the two roots of the equation from the unknown quantity.

295. A QUADRATIC EXPRESSION is a trinomial which contains the first and second powers of some letter or quantity.

The principle established in the last article furnishes a method of resolving any quadratic expression into simple binomial factors.

Assuming the quantity within the parenthesis equal to 0, and transposing, we obtain

3x2-10 x 25 = 3 (x — 5) (x + §).

-

3. Resolve x2

6x+8 into binomial factors.

Ans. (x4) (x2).

4. Resolve x2 + 73x + 780 into binomial factors.

Ans. (+60) (x+13).

Ans. 2(x+2) (x-3).

5. Resolve 2x2 + x - 6 into binomial factors.

6. Resolve x2 + 52 x — 256 into binomial factors.

Ans. (x-4) (x+4).

FORMATION OF EQUATIONS.

296. The principles established in Articles 293 and 294 furnish methods of readily forming a quadratic equation when its roots are given.

EXAMPLES.

1. Required the equation whose roots are 4 and 7.

2. Form the quadratic equation whose roots are 1 and Ans. x2+x= 2.

- 2.

3. Form the quadratic equation whose roots are 4 and 5. Ans. x2 9x= 20.

4. Form the quadratic equation whose roots are 1+√5 and 1-5. Ans. x2 2x 4.

5. Find the quadratic equation whose roots are 3 and 12x9.

Ans. 5 x2

6. Find the quadratic equation whose roots are 7 and

– 63.

7. Find the quadratic equation whose roots are m+ √n

DISCUSSION OF THE GENERAL EQUATION.

297. The values of p and q in the general equation may be either positive or negative. If those quantities be considered essentially positive, and the signs be expressed, we shall have four forms.