the absolute term (that is, the second side,) in the proposed by four times the first coefficient, adding to the result the square of the second coefficient, and covering the whole by the sign of the square root. As an illustration, take Example 1, Art. 91, which, after transposition, is

3x2+5x=42;

then, forming each side of the reduced equation as above directed, we get immediately

When the coefficients and absolute term in a quadratic equation are very large numbers, the solution may be more expeditiously obtained by the method explained in The Chapter on the General Theory and Solution of Equations of all Degrees, which forms a supplement to the present volume.

(93.) QUESTIONS PRODUCING QUADRATIC EQUATIONS INVOLVING

BUT ONE UNKNOWN QUANTITY.

QUESTION I.

It is required to find two numbers, whose difference shall be 12, and product 64.

Let x be the less number;

then x + 12 is the greater:

also by the question, x (x+12)= 64,

that is, x2+12x = 64;

.. completing the square, x2 + 12x + 36 = 100;

and extracting the root, x+6=±10;

.. x = ± 10 — 6 — 4, or — 16:

hence the numbers are either 4 and 16, or- - 16 and

- 4.

QUESTION II.

Having sold a commodity for 567., I gained as much per cent. as the whole cost me. How much then did it cost?

or, by transposition, +100x = 5600;

and completing the square, x2 + 100x + 2500 = 8100;

.. extracting the root, x + 50=90;

.. the commodity cost 407.: the other value of x is inadmissible.

QUESTION III.

A company at a tavern had 87. 15s. to pay; but, before the bill was paid, two of them went away, when those who remained had, in consequence, 10s. each more to pay. How many persons were company at first?

in

Let x be the number;

then, is the number of shillings each had to pay at first ;

175
x

and completing the square, x2-2x+1=36;

350+10x2 = 195x;

A person travels from a certain place at the rate of one mile the first day, two the second, and so on; and, in six days after, another sets out from the same place, in order to overtake him, and travels uniformly at the rate of fifteen miles a day. In how many days will they be together?

Let x be the number of days;

Then the first will have travelled x + 6 days;

and, .. x2 + 13x + 42 = 30x;

or, by transposition, x2 17x = 42;

and completing the square

(Art. 91), 4x2-68x + 289 = 121 ; -17= ±11;

.. extracting the root, 2x

hence it appears, that they will be together 3 days after the second sets out, who will then overtake the first, and be overtaken by him again in 11 days after, or 14 from the time of the second setting out.

QUESTION v.

A vintner sold 7 dozen of sherry and 12 dozen of claret for 50l., and finds that he has sold 3 dozen more of sherry for 107. than he has of claret for 61. Required the price of each.

Let x be the price of a dozen of sherry in pounds;

or 70x21x2 + 72x = 500-150x;

.. by transposition, 292x-212= 500;

or x2-3x=-500;

and completing the square, x2-
x2-203x + (44)2 =

... the price of a dozen of sherry was 21., and of a dozen of claret

6x

) 37. If the other value of x be admitted, then the

10 3x number expressing the dozens of claret will be negative: it is therefore inadmissible.

QUESTION VI.

It is required to find two numbers, such, that their sum, product, and difference of their squares, shall be all equal.

Let the numbers be represented by x and x + 1, then one condition will necessarily be fulfilled; for the sum, x + (x + 1), and the difference of the squares, (x + 1)2 - x2, are each 2x + 1. We have therefore only to satisfy the remaining condition, that is, to solve the equation

7. It is required to find two numbers, whose sum shall be 14, such, that 18 times the greater shall be equal to 4 times the square of the less.

Ans. 6 and 8.

8. Divide the number 48 into two such parts that their product may be 432.

Ans. 36 and 12.

9. Divide the number 24 into two such parts that their product may be equal to 35 times their difference.

Ans. 10 and 14.

10. What number is that which exceeds its square root by 484.

Ans. 564.

11. It is required to find two numbers, the first of which may be to the second as the second is to 16; and the sum of their squares equal to 225.

Ans. 9 and 12.

12. A person bought some sheep for 727., and found that if he had bought 6 more for the same money, he would have paid 17. less for each. How many did he buy, and what was the price of each?

Ans. The number of sheep was 18, and the price of each 47.

13. A merchant sold a quantity of brandy for 397., and gained as much per cent. as it cost him. What was the price of the brandy? Ans. 301.

14. In a parcel containing 24 coins of silver and copper, each silver coin is worth as many pence as there are copper coins; and each copper coin is worth as many pence as there are silver coins ; and the whole is worth 188. How many are there of each? Ans. 6 silver coins, and 18 copper coins;

or 18 silver, and 6 copper.

15. A traveller sets out for a certain place, and travels one mile the first day, two the second, three the third, and so on: in 5 days afterwards another sets out, and travels 12 miles a day. How long and how far must he travel to overtake the first?

Ans. He must travel 3 days, or 36 miles.