2d. We will not write out the transformed equations; that is, not attach the letters to the coefficients; we can then unite the whole in one operation.

3d. Consider the root a quotient; the absolute term a dividend, and, corresponding with these terms, we must have divisors.

In the example under consideration, 8 is the first divisor; 15 is the first trial divisor; 15.2 is the second divisor, and 15.4 is the second trial divisor; 15.46 is the third divisor, &c.

Let us now generalize the operation. The equation may be represented by x2+ax=n

Transform this into another whose roots shall be less by r that equation into another whose roots shall be less by s, &c., &c.

In the above we have represented the difference between n and (arr) by n', &c. As n', n", n"", &c., with their corresponding trial divisors, will give s, t, u, &c., the following for mula will represent the complete divisors for the solution of all equations in the form of

Equations which have expressed coefficients of the highest power, as

To obtain trial divisors we would add cr only, in place of (cr+cs), &c.

We will now resume our equation for a more concise solution.

We can now divide as in simple division, and annex the quotient figures to the root, thus:

2. Find x, from the equation x2-700x=59829.

On trial, we find x cannot exceed 800; therefore, 'r=700.

3. Find x from the equation x2-1283x-16848.

By trial, we find that a must be more than 1000, and less than 2000; therefore, 1000:

n rstu 16848(1296

4. Given a2-5x=8366, to find x.

By trial, we find x must be more than 90, and less than 100.

Therefore, at r =

85) 8366 (94=x

r+s.. 94 765

a+2r+s=179) 716

716

5. Find x, from the equation x2-375x+1904=0.

Here the first figure of the root is 5.

7. Given a

Ans. 31.2311099.

21x=214591760730, to find x, Ans. 463251.

It might be difficult for the pupil to decide the value of r, as applied to the last example, without a word of explanation: must be more than the square root of the absolute term, that is,

more than 400000; then try 500000, which will be found too great.

(Art. 192.) When the coefficient of the highest power is not unity, we may (if we prefer it to using the last formulas for di-visors), transform the equation into another, (Art. 166.), which shall have unity for the coefficient of the first term, and all the other coefficients whole numbers.

8. Given 7x2—3x=375.

y

Put = and we shall have y2—3y=2625.

One root of this equation is found to be 52.7567068+, oneseventh of which is 7.536672+; the approximate root of the original equation.

9. Given 7x2-83x+187=0, to find one value of x.

Ans. 3.024492664

10. Given x2-8, to find one value of x.

Ans. 2.96807600231

11. Given 4+, to find one value of x.

Ans. Ans. .14660+

12. Given 2+3x=777, to find one value of x.

13. Given 115-3x2-7x=0, to find one value of x.

(Art. 193.) We now apply the same principle of transformation to the solution of equations of the third degree.

EXAMPLES.

1. Find one root of the equation

x-x70x-300=0.

We find, by trial, that one root must be between 3 and 4.