mainders, we can obtain its approximate root, in decimals, to any assignable degree of accuracy. Two zeroes are annexed for each new decimal of the root, for the same reason that two figures are brought down, when the operation is confined to whole numbers. Of course, the decimal places of the root will be equal to half the number of zeroes used.

7 (2.645 &c.

4

46 300

276

524 2400

2096

5285 30400

26425

3975

If there were no remainder, the square root of 7 would be 26; but, as there is a remainder, it is more than 2, although less than 2. Therefore, differs less than the thousandth part of one from the true root required. By annexing additional zeroes, and continuing the work, we may obtain the root still more accurately.

2

2. Required the square root of 5.
3. What is the square root of 2?
4. Extract the square root of 823.
5. Find the square root of 527.

6. What is the square root of?

The square root of 9 is 3; but, since the numera

A

tor is not a complete power, the root whose second power is nearest to 5 must be taken, which is 2. The difference between and the true square root of

less than of a unit.

, is

When both the numerator and the denominator of a fraction are multiplied by the same quantity, its value is not altered; and if both terms of this fraction be multiplied by any perfect power, a nearer approximation to the true root will be obtained. multiplied by 9.

The square root of 81 root of 45 is 7; so that

Let them be

is 9, and the nearest square expresses the value of

Again, letbe multiplied by 144, which is the

within part of a unit.

In general, the larger the power by which the terms of a fraction are multiplied, the nearer to the true root will be the approximation.

7. What is the square root of 34?

Reduce this mixed number to an improper fraction,

The nearest square root of 15 is 2 or 14, which

is the root required within less than

8. Required the square root of g. 9. What is the square root of 5? 10. Find the square root of .

11. Required the square root of 23.

of one.

12. What is the square root of a2 + x2?

It has been shown, already, that no binomial is a perfect second power. The approximate root of a surd can be found by the common rule for extracting the square root of a compound quantity, thus:

13. Required the square root of 1+x.
14. What is the square root of x2 22?
15. Extract the square root of a1 + 1.
16. What is the square root of?
17. Extract the square root of 7641.

+

x12

256 alo

x12

256 a10

CHAPTER XI.

EQUATIONS OF THE SECOND DEGREE.

SECTION I.

Pure Equations.

An equation of the second degree contains the second power of the unknown quantity. When the unknown quantity appears only in the second power, the equation is said to be pure.

1. What number is that, which, being multiplied by itself, and the product doubled, will give 162? Let the number.

2. A farmer, being asked how many cows he had, answered, that if the number were multiplied by 5 times itself, the product would be 720. How many had he?

Let

the number of cows.

Then 5 x 720, by the question.

=

r2 = 144, and x = 12.

ANS. 12 COWS.

3. A gentleman, being asked the price of his hat, answered, that if it were multiplied by itself, and 26 were subtracted from the product, the remainder multiplied by 5 would be 190. What was the price of the hat?

Let x = the price of the hat.

Then 52130 190, by the question.
5 x2 = 190 +130, or 320, by transposition.
x264, and x = 8.

ANS. $8.

4. A gentleman, being asked the age of his son, replied, that if from the square of his age were subtracted his own age, which was 30 years, and the remainder were multiplied by his son's age, the product would be 6 times his age. How old was he?

Let x the son's age.

Then, by the conditions of the question,
(x2 - 30) x, or x3 30 x = 6 x.

2306, by dividing by x.

x2 = 36, and x = 6.

ANS. 6 years.

5. What two numbers are those, which are to each other as 3 to 4, and the difference of whose squares is 112?

Let the larger number,

ANS. 16 and 12.