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PRODUCING SIMULTANEOUS QUADRATIC EQUATIONS CONTAINING
TWO UNKNOWN NUMBERS.

289. Some of the following problems may be solved

with one unknown number.

negative values are omitted.

In the answers given, the

1. There is a certain number of two figures, whose product is 10; and if 56 is added to the number, and the sum of the squares of its figures subtracted from the sum, the order of the figures will be reversed. Find the number.

2. The area of a rectangular field is 975 square rods, and if the length were decreased 5 rods, and the breadth increased 5 rods, the area would be 1020 square rods. What are the length and breadth ?

Ans. Length, 39 rods; breadth, 25 rods.

3. The difference of the cubes of two numbers is 316, and the sum of their squares plus their product is 79. the numbers?

What are

4. The area of a rectangular field whose perimeter is 88 rods is 363 square rods. Find the length and breadth.

5. The fore wheels of a carriage make 8 more revolutions than the hind wheels in going 160 yards, but if the circumference of each wheel is increased 3 feet the carriage must pass over 240 yards in order that the fore wheels may make 8 revolutions more than the hind wheels. What is the circumference of the wheels?

Ans. Fore wheels, 12 feet; hind wheels, 15 feet.

The

6. There are two pieces of cloth of different lengths. difference of the squares of the number of yards in each is 76; and the product of the numbers representing the number of yards in each, plus one half the square of the number of yards in the longer is 560. Find the length of each.

7. Find two numbers such that 4 times the square of the greater minus 5 times the square of the less shall be 20, and 5 times the square of the less plus their product shall be 100.

8. A drover bought 10 oxen and 18 cows for $1040, buying one more ox for $150 than cows for $60. Find the price of an ox, and the price of a cow. Ans. Ox, $50; cow, $30.

9. Find two numbers such that their sum, their product, and the sum of their squares shall be equal to one another.

10. A and B, talking of their ages, find that the square of A's age minus the product of the ages of both is 385, and

3 times this product plus the square of B's age is 3096. Find the age of each.

11. A and B purchased a wood-lot containing 100 acres, each agreeing to pay $5000. Before paying for the lot, A offered to pay $10 an acre more than B, if B would allow A to have his choice in the division of the lot. How many acres should each receive, and at what price an acre?

Ans. S A, 47.5 acres, at $105.25— an acre. 52.5 acres, at $95.25+ an acre.

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12. Two sums of money, amounting to $9000, are loaned at such a rate of interest that the income from each is the same. But if the first part had been at the same rate as the second, the income from it would have been $160; while if the second had been at the same rate as the first, the income from it would have been $250. Find the rate of each.

Ans. 1st, 5%; 2d, 4%.

13. A boat's crew, rowing at half their usual rate, row 4 miles down a river and back in 3 hours and 20 minutes. At their usual rate they can go over the same course in 1 hour and 20 minutes. Find the usual rate of the crew, and the rate of the current.

14. A, working alone, built 8 rods of wall; then he hired B to work with him, and at the end of 3 days from the time A began they had completed 24 rods. Again, A worked 3 days and B 1 day, building 21.6 rods. Find the number of rods each can build a day, and the number of days B worked.

15. A reservoir is filled in 10 hours by means of several pipes, through which the water flows at a uniform rate. If there were 2 less pipes, and each pipe discharged 50 gallons more an hour, the reservoir would be filled in 12 hours and 30 minutes. If there were 3 more pipes, and each pipe discharged 25 gallons less an hour, the reservoir would be filled in 7 hours and 30 minutes. Find the number of pipes, and the capacity of the reservoir.

16. A tailor bought two pieces of cloth for $100. For the first he paid as many dollars a yard as there were yards in both; and for the second as many dollars a yard as there were yards in the first more than in the second; and the first piece cost 3 times as much as the second. Find the number of yards, and the cost a yard of each.

17. A man walks 3 hours at the rate of 3 miles an hour, and then takes a different rate. After a number of hours he finds that, if he had kept his first rate, he would have been 2 miles less distant from his starting point, while if he had walked at his first rate 2 hours, and at his second rate 5 hours, he would be half a mile farther from his starting point. Find the whole time of the walk, and the entire distance travelled.

Ans. 7 hours, and 23 miles.

18. The soldiers of a regiment can be arranged so as to have twice as many men in a line as there are lines. But 16 men must be added to the regiment in order that the men may be arranged in a hollow square six deep, having the same number of men in each outer side of the square as there were in the lines before. Find the number of men in the regiment.

19. The area of a certain rectangle is equal to the area of a square whose side is 6 meters less than one of the sides of the rectangle, and also equal to the area of another rectangle whose length is 2 meters less, and width 1 meter more. Find the length of the sides of the rectangle.

20. At 7 o'clock A. M., A and B set out in opposite directions on their bicycles, from the same point. A's hourly rate was 5 miles. B, riding at a fixed rate, after a while turned and followed A. Three hours after he turned, B passed the point where A was when B turned, and at 12 o'clock he had reduced the distance between them at the time of turning one half. Find B's rate, the time when he turned, the distance between A and B at that time, and the time when B will overtake A if both continue at the same rate of speed.

CHAPTER XXI.

PROPERTIES OF QUADRATIC EQUATIONS.

290. EVERY affected quadratic equation can be reduced (§ 274) to the form

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Thus, there are two roots of a quadratic equation, and the sum of

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That is, the sum of the roots is equal to the coefficient of the second term with its sign changed; the product of the roots is equal to the last term.

291. A quadratic equation, x2 + bx + c = 0, will have two real and unequal roots, two real but equal roots, or two imaginary roots, according as b2 is >, =, or < 4 c.

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Now if b2 4 c, the expression under the radical is positive, and the roots are real and unequal.

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If b2 <4 c, the expression under the radical is negative, and both roots are imaginary (§ 213).

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