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PROBLEMS PRODUCING QUADRATIC EQUATIONS.

Prob. 8. Find two numbers such that the product of then sum and difference may be 5, and the product of the sum of their squares and the difference of their squares may be 65.

Prob. 9. Find two numbers such that the product of their sum and difference may be a, and the product of the sum of their squares and the difference of their squares may be ma.

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Prob. 10. A laborer dug two trenches, whose united length was 26 yards, for 356 shillings, and the digging of each of them cost as many shillings per yard as there were yards in its length. What was the length of each?

Ans. 10, or 16 yards.

Prob. 11. What two numbers are those whose sum is 2a, and the sum of their squares is 2b ?

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Prob. 12. A farmer bought a number of sheep for 80 dollars, and if he had bought four more for the same money, he would have paid one dollar less for each. How many did he buy? Let x represent the number of sheep.

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x

would be the price of each, if he had bought four

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Prob. 13. A person bought a number of articles for a dollars. If he had bought 2b more for the same money, he would have paid c dollars less for each. How many did he buy?

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Prob. 14. It is required to find three numbers such that the

product of the first and second may be 15, the product of the first and third 21, and the sum of the squares of the second and third 74.

Ans. 3, 5, and 7.

Prob. 15. It is required to find three numbers such that the product of the first and second may be a, the product of the first and third b, and the sum of the squares of the second and third c.

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Prob. 16. The sum of two numbers is 16, and the sum of their cubes 1072. What are those numbers?

Ans. 7 and 9.

Prob. 17. The sum of two numbers is 2a, and the sum of their cubes is 26. What are the numbers?

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Prob. 18. Two magnets, whose powers of attraction are as 4 to 9, are placed at a distance of 20 inches from each other. It is required to find, on the line which joins their centers, the point where a needle would be equally attracted by both, admitting that the intensity of magnetic attraction varies inversely as the square of the distance.

8 inches from the weakest magnet,

Ans. {or-40 inches from the weakest magnet.

Prob. 19. Two magnets, whose powers are as m to n, are placed at a distance of a feet from each other. It is required to find, on the line which joins their centers, the point which is equally attracted by both.

Ans.

The distance from the magnet m is

The distance from the magnet n is

a√m √m ± √ n

±a√n √m±√n

Prob. 20. A set out from C toward D, and traveled 6 miles an hour After he had gone 45 miles, B set out from D toward C, and went every hour of the entire distance; and after he had traveled as many hours as he went miles in one

hour, he met A. Required the distance between the places C and D.

Ans. Either 100 miles, or 180 miles.

Prob. 21. A set out from C toward D, and traveled a miles per hour. After he had gone b miles, B set out from D toward C, and went every hour th of the entire distance; and after he had traveled as many hours as he went miles in one hour, he met A. Required the distance between the places C and D.

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Prob. 22. By selling my horse for 24 dollars, I lose as much per cent. as the horse cost me. What was the first cost of the horse?

Ans. 40, or 60 dollars.

QUADRATIC EQUATIONS CONTAINING TWO UNKNOWN QUAN

TITIES.

(186.) An equation containing two unknown quantities is said to be of the second degree when it involves terms in which the sum of the exponents of the unknown quantities is equal to 2, but never exceeds 2. Thus,

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are equations of the second degree.

The solution of two equations of the second degree containing two unknown quantities, generally involves the solution of an equation of the fourth degree containing one unknown quantity. Hence the principles hitherto established are not sufficient to enable us to solve all equations of this description. Yet there are particular cases in which they may be reduced either to pure or affected quadratics, and the roots determined in the ordinary manner.

(187.) When one of the equations is a simple equation, it is generally best to find an expression for the value of one of the unknown quantities from the simple equation, and substitute this value in the place of its equal in the other equation. The resulting equation will be of the second degree, and may be solved by the ordinary rules.

Ex. 1. Given x2+3xy— y2=23) to find the values of x+2y= 7) and y.

From the second equation, we find

Whence

x=7-2y.

x2-49-28y+4y2.

And, substituting this value in the first equation, we have

49-28y+4y+21y-6y'-y'=23,

a common quadratic equation, which may be solved in the usual manner.

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(188.) When the same algebraic expression is involved to different powers, it is sometimes best to regard this expression as the unknown quantity.

Ex. 4. Given x2+2xy+y2 +2x=120-2y to find the val

xy-y1=8

}

ues of x and y.

Here the first equation may be put under the form

(x+y)2+2(x+y)=120,

where x+y may be regarded as a single quantity, and, by completing the square, we shall find its value to be

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Proceeding now as in the last Article, we shall find

x=6, or 9, or -9√5,

y=4, or 1, or -35.

Ex. 5. Given 4xy=96—x'y' } to find the values of x and y.

x+y=6

Here we may regard xy as the unknown quantity, and we shall find its value from the first equation to be

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Proceeding again as in the former Article, we shall find

Ex. 6. Given

x

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Here may be treated as the unknown quantity, and we

y

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(189.) When the sum of the dimensions of the unknown quantities is the same in every term of the two equations, it is sometimes best to substitute for one of the unknown quantities the product of the other by a third unknown quantity.

Ex. 7. Given x2+xy =56

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xy+2y=60} to find the values of x and y.

Here, if we assume x=vy, we shall have

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From which, after completing the square, we obtain

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Substituting either of these values in one of the preceding expressions for y', we shall obtain the values of y; and since zvy, we may easily obtain the values of x.

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