CHAPTER XI.

ON

THE SOLUTION OF PROBLEMS,

PRODUCING QUADRATIC EQUATIONS.

§ 1. SOLUTION OF PROBLEMS PRODUCING QUADRATIC EQUATIONS, INVOLVING ONLY ONE UNKNOWN QUANTITY.

428. It may be observed, that, in the solution of problems which involve quadratic equations, we sometimes deduce, from the algebraical process, answers which do not correspond with the conditions. The reason seems to be, that the algebraical expression is more general than the common language, and the equation which is a proper representation of the conditions, will express other conditions, and answer other suppositions.

Prob. 1. A person bought a certain number of oxen for 80 guineas, and if he had bought four more for the same sum, they would have cost a guinea a piece less; required the number of oxen and price of each.

Let the number; then

30

the price of each;

х

and by reduction, x+4x=320;

•'. x2+4x+4=324, and x+2=±18;

80

And

=

80

16

..x=16, or -20,

=5 guineas, the price of each.

The negative value (—20) of x, will not answer the condition of the problem.

Prob. 2. There are two numbers whose difference is 9, and their sum multiplied by the greater produces 266. What are those numbers?

Let the greater; .. x-y= the less. x

and x.(2x-9)=266; .'. x2

completing the square, &c. x

9 2

266

.x=

2

..x=14, or -91; and x-9=5, or -18. Here both values answer the conditions of the problem.

Prob. 3. A set out from C towards D, and travelled 7 miles a day. After he had gone 32 miles, B set out from D towards C, and went every day one-nineteenth of the whole journey; and after he had travelled as many days as he went miles in one day, he met A. Required the distance of the places C and D.

Suppose the distance was x miles.

х the number of miles B travelled per day;

19

19

30

19

extracting the root, -6=±2;

8, or 4; and x=152, or 76, both which values answer the conditions of the problem. The distance therefore of C from D was 152, or 76 miles.

Prob. 4. To divide the number 30 into two such parts, that their product may be equal to eight times their difference.

Let x the lesser part; .. 30-x- the greater part, and 30-x-x, or 30-2x= their difference. Hence, by the problem, x(30—x)=8(30—2x), or 30x-x2=240-16x; .. x2-46x=-240.

Completing the square, x2-46x+529=289; ..x=23±17=40, or 6= lesser part;

and 30—x=30-6=24= greater part.

In this case, the solution of the equation gives 40 and 6 for the lesser part. Now as 40 cannot possibly be a part of 30, we take 6 for the lesser part, which gives 24 for the greater part; and the two numbers, 24 and 6, answer the conditions required.

Prob. 5. Some bees had alighted upon a tree; at one flight the square root of half of them went away; at another eight-ninths of them; two bees then remained. How many then alighted on the tree?

16x3
9

Let 2x2=the number of bees; x+ +2=2x2,

or 9x+16x+18=18x .. 2x2-9x=18; (Art. 417), Multiplying by 8, 16x3-72x=144 ; adding 81 to both sides, 16x2-72x+1=225 5 ;. ..4x=9±15=24, or 5; and x=6, or -1. .'. 2x2=72, or4. But the negative value 13 of a, is excluded by the nature of the problem; therefore, 72= number of bees.

429. If, in a problem proposed to be solved, there are two quantities sought, whose sum, or difference, is equal to a given quantity, for instance, 2a; let half their difference, or half their sum, be denoted by x; then x+a will represent the greater, and x-a the lesser, (Art. 102). According to this method of notation, the calculation will be greatly abridged, and the solution of the problem will often be rendered very simple.

Prob. 6. The sum of two numbers is 6, and the sum of their 4th powers is 272. What are the numbers?

Let x half the difference of the two numbers; then 3+x= the greater number, and 3-x the lesser.

.. by the problem, (3+x)+(3-x)*=272, or 162+108x+2x=272; from which, by transposition and division, x4 +54x2=55;

... completing the square, x4 +54x2+729=784, and extracting the root, x2+27=±28; ..x2=27±28, and a=±l, or ±√-55.

Now, by taking the positive valué, +1, for x, (since in this case, it is the only value of x which will answer the problem); we shall have 3+1=4= the greater, and 3-1-2= the lesser.

Prob. 7. To divide the number 56 into two such parts, that their product shall be 640.

Ans. 40, and 16. Prob. 8. There are two numbers whose difference is 7, and half their product plus 30, is equal to the square of the lesser number. What are the numbers? Ans. 12, and 19.

Prob. 9. A and B set out at the same time to a place at the distance of 150 miles. A travelled 3 miles an hour faster than B, and arrives at his journey's end 8 hours and 20 minutes before him. what rate did each person travel per hour?

At

Ans. A 9, and B 6 miles an hour. Prob. 10. The difference of two numbers is 6; and if 47 be added to twice the square of the lesser, it will be equal to the square of the greater. What are the numbers? Ans. 17, and 11.

Prob. 11. There are two numbers whose product is 120, if 2 be added to the lesser, and 3 subtracted from the greater, the product of the sum and remainder will also be 120. What are the numbers?

Ans. 15, and 8.

Prob. 12. A person bought a certain number of sheep for 1201. If there had have been 8 more, each would have cost him ten shillings less. How many sheep were there? Ans. 40.

Prob. 13. A Merchant sold a quantity of brandy for 391. and gained as much per cent as the brandy cost him. What was the price of the brandy?

Ans. 301.

Prob. 14. Two partners, A and B, gained 187. by trade. A's money was in trade 12 months, and he received for his principal and gain 261. Also, B's money, which was 301, was in trade 16 months. What money did A put into trade? Ans. 201.

Prob. 15. A and B set out from two towns which were at the distance of 247 miles, and travelled the

direct road till they met. A went 9 miles a day; and the number of days, at the end of which they met, was greater by 3 than the number of miles which B went in a day.. How many miles did each go?

Ans. A 117, and B 130 miles.

Prob. 16. A man playing at hazard won as the first throw, as much money as he had in his pocket; at the second throw, he won 5 shillings more than the square root of what he then had; at the third throw, he won the square of all he then had; and then he had 1127. 16s. What had he at first?

Ans. 18 shillings.

Prob. 17. If the square root of a certain number be taken from 40, and the square root of this difference increased by 10, and the sum multiplied by 2, and the product divided by the number itself, the quotient will be 4. Required the number.

Ans. 6.

Prob. 18. There is a field in the form of a rectangular parallelogram, whose length exceeds the breadth by 16 yards; and it contains 960 square yards. Required the length and breadth. Ans. 40 and 24 yards.

Prob. 19. A person being asked his age, answered, if you add the square root of it to half of it, and subtract 12, there will remain nothing. Required his age. Ans. 16.

Prob. 20. To find a number, from the cube of which, if 19 be subtracted, and the remainder multiplied by that cube, the product shall be 216.

Ans. 3, or -2. Prob. 21. To find a number from the double of which if you subtract 12, the square of the remainder, minus 1, will be 9 times the number sought.

Ans. 11, or 31.

Prob. 22. It is required to divide 20 into two such parts, that the product of the whole and one of the parts, shall be equal to the square of the other.

Ans. 10/5-10, and 30-10/5.

Prob. 23. A labourer dug two trenches, one of which was 6 yards longer than the other, for 177.