18. A man had a field, the length of which exceeded the breadth by 5 rods. He gave 3 dollars a rod to have it fenced, which amounted to 1 dollar for every square rod in the field. What was the length and breadth, and what did he give for fencing it?

i

19. From two places at a distance of 320 miles, two persons, A and B, set out at the same time to meet each other. A travelled 8 miles a day more than B, and the number of days in which they met was equal to half the number of miles B went in a day. How many miles did each travel, and how far per day?

20. A man has a field 15 rods long and 12 rods wide, which he wishes to enlarge so that it may contain just twice as much ; and that the length and breadth may be in the same proportion. How much must each be increased?

In this example, the root can be obtained only by approx mation.

21. A square court yard has a rectangular gravel walk round it. The side of the court wants 2 yards of being 6 times the breadth of the gravel walk; and the number of square yards in the walk exceeds the number of yards in the periphery of the court by 164. Required the area of the court?

All equations of the second degree may be reduced to one of the following forms.

After the equation has been brought to one of these forms, it may be solved by one of the following formulas, which are numbered to correspond to the equations from which they are derived.

The first equation and the first formula are sufficient for the whole, if p and q are supposed to be positive or negative quantities.

21. There are two numbers whose difference is 113, and whose product is equal to 4 times the larger minus 9. What are the numbers?

This equation is in the form of x2- p. x = — q,

q, in which

x = 78 ± (%%!% −9)3 = 78 ± (518)1 = 7.8±7.2.

Or we may use the first formula, then

= 78 ± (!!!) + = 7.8±7.2.

Both values of x, being positive, will answer the conditions of the question.

Ans. By the first value the larger number is 15 and the smaller 33. By the second value of x, the larger is 3, and the smaller 11.

Let the learner solve some of the preceding questions by the formula.

XXXV. We shall now demonstrate that every equation of the second degree, necessarily admits of two values for the unknown quantity, and only two.

This, we have seen, may represent any equation whatever of the second degree, p and q being any known quantities and either positive or negative. If p = 0 the equation becomes

which is a pure equation or an equation with two terms.

If we make the first member of the equation x2 +px=g, a complete second power, by the above rules, it becomes

transposing m2 (x+2)'— m2 = 0.

The first member of this equation is the difference of two second powers, which, Art. XIII, is the same as the product of the sum and difference of the numbers.

The sum is x+2+m, and the difference is x + 2

and their product is

(x+2−m) (x+2+m) = 0.

m,

In this equation, the first member consists of two factors, and the second is zero. Now the first member of the above equation will be equal to zero, if either of its factors is equal

to zero. For if any number be multiplied by zero, the product is zero.

Making the first factor equal to zero,

Either of these values of x must answer the conditions of the equation.

N. B. Though either value answers the conditions separately, they cannot be introduced together, for being different, their product cannot be x2.

Instead of m put its value, and the values of x become

which are the values we had obtained above. (This demonstration is essentially that of M. Bourdon.) ·

Discussion.

Let us take again the general equation.

x = − 2 = ( 1 + 2 ) + .

Since the expression contains a radical quantity, that is, a quantity of which the root is to be found, in order to be able to find the value of it, we must be able to find the root either exactly or by approximation. Now there is one case in which

it is impossible to find the root. It is when q is negative and greater than 2. In which case the expression q +2

4

is ne

gative; and it has been shown above, that it is impossible to find the root of a negative quantity. In all other cases the value of the equation may be found.

In all cases if q is positive, the first value will be positive, and answer directly to the conditions of the question proposed.

For the radical (2+2) is necessarily greater than

P, be

2

cause the root of alone is 2 ; therefore the expression

4

- 2 + (2+2) is necessarily of the same sign as the

radical.

The second value is for the same reason essentially negative,

for both and (4+2) are negative. This value, though

2

it fulfils the conditions of the equation, does not answer the conditions of the question, from which the equation was derived; but it belongs to an analogous question, in which the x must be put in with the sign instead of +; thus x2 -px=q, which

gives x = // ± (2 + 2), a value, which differs from the

2

(༡

In order that it may be possible to find the root, g must be

less than

When this is the case, the two values are real.