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2a3-3a2b

aa—6a°b+15 a*b° — 20a3b'+ 15 a°b'—6ab3 +b° ( a2;

ασ

−6a3b+15a4b
-6ab9a+b2-3a2b, second term.

2 a3 — 6a2b+3ab2) 6a+ b2 —20a3b3 +15a°b*

a3, first term.

6ab18ab+b+3ab, third term.

— a3 b3 9a2b1

2a3b3 + 6a2ba — 6 a b3 +b
2ab+6a2b4-6ab5+b

2 a3 — 6a2b+6ab2—b3 )=

[blocks in formation]

-6,4th term.

a3-3a2b+3ab-b3, root required.

189. From the rule that we have thus explained we may easily deduce the method that is usually given in books of arithmetic for the extraction of the square root, as will be seen from the following examples in numbers.

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190. If, after the whole operation is finished, there be a remainder, we must conclude that the number proposed is not a square, and that consequently we are unable to determine its root. In that case we can only make use of the radical sign which we have before employed. Thus the root of a2+b2 must be represented by (a2+b2), or √a2+b2.

CHAP. XX.

OF ALGEBRAIC EQUATIONS.

191. THE principal object of Algebra, as well as of all other branches of mathematics, is to determine the value of quantities which were before unknown, and this is done by an attentive examination of the conditions given. For this reason Algebra has been defined, The science which teaches how to determine unknown quantities by means of those that are known.

This definition agrees with all that has been hitherto laid down; for we have seen that the knowledge of certain quantities has in all cases led to the determination of others, that had been before regarded as unknown.

Addition furnishes us with an example of this; for, in order to find the sum of two or more given numbers, we had to seek for an unknown number, which should be equal to the known numbers taken together. In Subtraction also we had to find a number equal to the difference of two known numbers.

A multitude of other examples present themselves in Multiplication and Division; in the raising of powers, and the extraction of roots. The question is always reduced to finding, by means of known quantities, others that were before undetermined; or, in other words, to finding, by the aid of certain given numbers, a new number, which should bear to them a certain relation, and this relation is determined by certain conditions or properties, which were to agree with the quantity sought.

192. Now when it is proposed to resolve a question of this nature, we represent the unknown quantity, or number

sought,

sought, by one of the last letters of the alphabet, and proceed to examine in what manner the conditions given can form an equality between two quantities; this equality is represented by a certain formula, called an equation, by means of which we are enabled to determine the value of the number sought, and consequently to resolve the question. It sometimes happens that more than one number is sought, but they are determined in the same manner by equations.

In order to illustrate this by the simplest example, let us propose the following question, or problem: What number is that, which, if 8 be added to it, will amount, or be equal to 15?

Now this number being for the present unknown, let it be represented by the letter x, and with this letter let us proceed in the same manner as if the number itself were known, and we were desirous of trying whether it corresponded with the conditions of the question. We shall then have the following equation;

x+8=15;

and by subtracting 8 from each side of the equation, there arises another, viz.

x=7;

from which the value of x is determined, and a known number found, answering to the conditions of the question.

198. We thus perceive that an equation consists of two parts, separated by the sign of equality,, to show that those two parts are equal to each other. It is often necessary to subject those parts to a great number of transformations, in order to deduce the value of the unknown quantity; but these transformations must be founded in all cases upon the following theorems; viz. that

Two equal quantities will remain equal, whether we add to, or subtract from them equal quantities; whether we multiply, or divide them by the same number; whether we raise them

both

both to the same power, or extract their roots of the same degree.

The equations, which are most easily resolved are those in which the unknown quantity does not exceed the first power, after the terms of the equation have been properly arranged; and these equations are called simple equations, or equations of the first degree. These equations are of two kinds; viz. those that involve only one unknown quantity, and those which involve two or more. When the equation contains the square of the unknown quantity, it is called a quadratic equation, or an equation of the second degree. Cubic equations, or equations of the third degree, are those which contain the cube of the unknown quantity, &c.

CHAP. XXI.

OF SIMPLE EQUATIONS INVOLVING ONLY ONE UNKNOWN QUANTITY.

194. WHEN the number sought, or unknown quantity, is represented by the letter x, and the equation is such that one side of it contains only x, and the other simply a known number, as x=25, the value of x is already determined. At this form, therefore, we must always endeavour to arrive, however complicated the equation may be, when first proposed; and it is now necessary to consider those rules, which will render the reduction of equations to this form the more easy.

195. In the example given in the preceding chapter, x+8=15,

we found immediately x=7.

And in general in the equation

x+a=b,

where

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