104. In order to approximate the square root of a fraction, the method, which first presents itself, is, to extract, by approximation, the square root of the numerator and that of the denominator; but with a little attention it will be seen, that we may avoid one of these operations by making the denominator a perfect square. This is done by multiplying the two terms of the proposed fraction by the denominator. If it were required, for example, to extract the square root of, we might change this fraction into

by multiplying its two terms by the denominator, 7. Taking the root of the greatest square contained in the numerator of this fraction, we have for the root of, accurate to within 4.

If a greater degree of exactness were required, the fraction must be changed by approximation or otherwise into another, the denominator of which is the square of a greater number than 7. We shall have, for example, the root sought within, if we convert into 225ths, since 225 is the square of 15; thus the fraction becomes 675 of one 225th, or, within; the root of 22 falls between and g, but approaches nearer to the second fraction than to the first, because 96 approaches nearer to a hundred than to 81; we have then 1 or for the root of within 5.

9 6

91

By employing decimals in approximating the root of the numerator of the fraction, we obtain 4,583 for the approximate root of the numerator 21, which is to be divided by the root of the new denominator. The quotient thence arising, carried to three places of decimals, becomes 0,655.

105. We are now prepared to resolve all equations involving only the second power of the unknown quantity connected with known quantities.

We have only to collect into one member all the terms containing this power, to free it from the quantities, by which it is multiplied (11); we then obtain the value of the unknown quantity by extracting the square root of each member.

Let there be, for example, the equation

Making the divisors to disappear, we find first

15x2 168 84 - 14x2.

Transposing to the first member the term 14 x2, and to the second the term 168, we have

It should be carefully observed, that to denote the root of the fraction 25, the sign is made to descend below the line, which seperates the numerator from the denominator. If it were

written thus,

252 29

the expression would designate the quotient

arising from the square root of the number 252 divided by 29; a result different from, which denotes, that the division is to be performed before the root is extracted.

29

I would remark here, that in order to designate the square root of a compound quantity, the upper line must be extended over the whole radical quantity.

The root of the quantity 4 a2b-263+3 is written thus,

√(4 a2 b − 2 b3 + c3),

by substituting, for the line extended over the radical quantity, a parenthesis including all the parts of the quantity, the root of which is required. This last expression may often appear preferable to the other (35).

In general, every equation of the second degree of the kind we are here considering, may, by a transposition of its terms, be reduced to the form

p x2

P designating the coefficient, whatever it may be, of x. We then

obtain

106. With respect to numbers taken independently, this solution is complete, since it is reduced to an operation upon the num

aq

ber either entire or fractional, which the quantity represents, an

P

arithmetical operation leading always to an exact result, or to one, which approaches the truth very nearly. But in regard to the signs, with which the quantities may be affected, there remains, after the square root is extracted, an ambiguity, in consequence of which every equation of the second degree admits of two solutions, while those of the first degree admit of only one.

Thus in the general equation a2 = 25, the value of x, being the quantity, which, raised to its square, will produce 25, may, if we consider the quantities algebraically, be affected either with the sign + or for whether we take +5, or 5, for this value,

.;

we have for the square

+5x+5= + 25, or

we may therefore take

or

in which the double sign shows, that the numerical value of

From what has been said, we deduce the general rule, that the double sign is to be considered as affecting the square root of every quantity whatever.

It may be here asked, why x, as it is the square root of 2, is

not also affected with the double sign? We may answer, first, that the letter x, having been taken without a sign, that is, with the sign, as the representative of the unknown quantity, it is its value when in this state, which is the subject of inquiry; and that, when we seek a number x, the square of which is b, for example, there can be only two possible solutions; x+√ō, x = −√b. Again, if in resolving the equation ab, we write ±x=±√b, and arrange these expressions in all the different ways, of which they are capable, namely,

=

we come to no new result, since by transposing all the terms of the equations x = + √ō, or which is the same thing, by changing all the signs (57), these equations become identical with the first.

107. It follows from the nature of the signs, that if the second member of the general equation

were a negative number, the equation would be absurd, since the square of a quantity affected either with the sign + or, having always the sign+, no quantity, the square of which is negative, can be found either among positive or negative quantities.

This is what is to be understood, when we say, that the root of a negative quantity is imaginary.

If we were to meet with the equation

we might deduce from it

or

but, there is no number,

16; but

which, multiplied by itself, will produce 16. It is true, that 4 multiplied by + 4, gives as these two quantities have different signs, they cannot be considered as equal, and consequently their product is not a square. This species of contradiction, which will be more fully considered hereafter, must be carefully distinguished from that mentioned in art. 58., which disappears by simply changing the sign of the unknown quantity; here it is the sign of the square x2, which is to be changed.

108. To be complete, an equation of the second degree, with only one unknown quantity, must have three kinds of terms, namely,

those involving the square of the unknown quantity, others containing the unknown quantity of the first degree, and lastly, such as comprehend only known quantities. The following equations are of this kind;

The first is, in some respects, more simple than the second, because it contains only three terms, and the square of x is positive, and has only unity for a coefficient. It is to this last form, that we are always to reduce equations of the second degree, before resolving them; they may then be represented by this formula, x2 + px = 9,

in which p and q denote known quantities, either positive or negaative.

It is evident, that we may reduce all equations of the second degree to this state, 1. by collecting into one member all the terms involving x (10), 2. by changing the sign of each term of the equation, in order to render that of x2 positive, if it was before negative (57), 3. by dividing all the terms of the equation by the multiplier of x, if this square have a multiplier (11), or by multiplying by its divisor, if it be divided by any number (12).

If we apply what has just been said to the equation

we have, by collecting into the first member all the terms involving x,

by changing the signs,

3x2+6x= 4,

If we now compare this equation with the general formula

109. In order to arrive at the solution of equations thus prepared, we should keep in mind what has been already observed (34), namely, that the square of a quantity, composed of two terms, always contains the square of the first term, double the product of the first term multiplied by the second, and the square of