SECTION XXXIX.

PROBLEMS PRODUCING AFFECTED QUADRATIC EQUATIONS.

310.-Ex. 1. What two numbers are such that their difference is 12, and their product 64?

Completing the square, x2+12x+62 = 64+36= 100

Extracting the square root,

whence,

and,

x+6 +10

Hence, the two numbers are 4 and 16, or 16 and -4, as either pair of values satisfies the problem.

2. What two numbers are such that their sum is 15, and their product 54 ?

Ans. 9 and 6.

3. What is that number from the square of which if we take 7 times the number, the remainder will be 44?

Ans. 11, or-4.

4. The ages of a man and his wife amount to 42 years, and the product of the numbers expressing their ages is 432. What is the age of each? Ans. The man, 24 years; the wife, 18.

5. A wholesale shoemaker received an unexpected order for 990 pairs of shoes, to be finished by the end of the month. He found that if these were divided equally among his men, each would have allotted to him 12 pairs more than he could get finished by the given time at his ordinary rate of working. He therefore engaged 54 more men, and got the work executed by the time specified. How many men had he at first?

6. A merchant sold a quantity of flour for $39, and gained as many per cent. as equalled the number of dollars in the price of the flour. What was the price of the flour?

Completing the square, x2+100x+(50)2=3900+2500=6400

Extracting the square root,

whence,

x+50=±80€

x=30, or -130

The cost of the flour was $30. The other value of x, not satisfying the conditions of the question, is not admissible.

7. A person invested a certain sum of money for goods, which he sold again for $24, and thereby lost as many per cent. as equalled the number of dollars invested. How much did he invest? Ans. $40, or $60.

8. There is a rectangular field whose length exceeds its breadth by 20 rods, and its area is 6300 square rods. What are its length and breadth?

9. A man planted a rectangular field with 8400 trees at equal distances, having 50 trees more in the longer rows than in the shorter ones. What was the number of trees in each of the

longer rows?

Ans. 120.

10. A trader computes that, during the time he has been in business, he has made $6300 clear profit. His neighbor, however, who has been three years less time in business, has made the same sum, owing to his clearing $27 per annum How long is it since the first commenced business?

more.

11. A farmer sold a number of tons of hay for $112, and observed that if he had sold one ton more for the same money, each ton would have brought him $2 less. Required the number of tons sold and the price per ton.

Completing the square, 4x2+4x+1=224+1=225

The number of tons sold was 7, and the price per ton was $16. The negative value of x is not admissible.

12. The sum of $144 was divided equally among a certain number of persons. If there would have received $1 more.

had been two persons less, each How many persons were there? Ans. 18.

13. A man bought a certain number of sheep for $80. If he had bought 4 more sheep for the same money, they would each have cost him $1 less. How many sheep did he buy?

14. Among a certain number of poor persons 110 bushels of coals were equally divided. If each person had received 1 bushel more, he would have received as many bushels as there were persons. Required the number of persons. Ans. 11.

15. A cistern is supplied with water by two pipes; by one of them it can be filled in 6 hours sooner than by the other, and by both together in 4 hours. Find the time in which each pipe will fill it.

SOLUTION.

x= the number of hours in which one will fill it; x+6=the number of hours in which the other will fill it.

The first will fill it in 6 hours, and the second in 12 hours. The negative value of x is not admissible.

16. A and B can perform a certain piece of work in 14% days, and A alone can perform it in 12 days less than B alone. Find the time in which A alone can perform it.

17. Two travellers, wishing to meet, set out from two towns, A and B, which are 120 miles distant from each other; the first goes 6 miles a day, and the other 1 mile a day more than the number of days in which they meet. In how many days will they meet?

Ans. 8.

18. The length of a rectangle exceeds its breadth by 12, and the sum of the squares of the length and breadth is 20880. What are the sides of the rectangle and the area?

Ans. Breadth, 96; length, 108; area, 10368.

19. In a concert-room 800 persons are seated on benches of equal length. If there were 20 fewer benches, it would be necessary that two more persons should sit on each bench. Find the number of benches.

Rejecting the negative value, we have 100 as the number of benches.

20. Divide 10 into two parts, such that the product of the parts shall be 24.

21. Two detachments of infantry are ordered to a station which is 39 miles distant. They begin their march at the same time; but one party, by travelling one-fourth of a mile an hour more than the other, arrives one hour sooner. Required the rates of marching per hour. Ans. 3 and 3 miles.

22. A had 40 yards of cloth, and B 90 yards, which they sold together for $42. Now, A sold for $1, a third of a yard more than B sold for the same money. How many yards did

each sell for $1?

23. The difference of two numbers is 2, and the difference of their cubes is 152. What are the numbers?

24. Two boys, John and William, start at the same time to walk a distance of 75 miles; but John walks 14 miles per hour faster than William, and finishes his journey 8 hours before him. How many miles per hour did each walk?

Ans. John, 4; William, 3.

SECTION XL.

SIMULTANEOUS QUADRATIC EQUATIONS.

311. A Homogeneous Equation is one in which the sum of the exponents of the unknown quantities in each term containing such quantities is the same.

Thus, x2 — y2 = 12 and x2 —xy+y2 = 12 are each homogeneous.

312. A Symmetrical Equation is one in which the unknown quantities are similarly involved.

Thus, x2+y2 = 25 and x2y-xy2 = 6 are each symmetrical.

313. Simultaneous Quadratic Equations (Art. 211) containing two unknown quantities cannot all be solved by the rules for quadratics.

The cases which can be treated in an elementary work are necessarily only such as can be solved by comparatively simple processes.