544 ALGEBRA [EQUATIONS IN GENERAL, found and increased by 1, the square root of the sura shall be equal to the given number diminished by 1. Let x be the number, then
- / 4 x i = x i - 2x, whence x = 0, x = - ,
neither of which solves the problem as stated. (3.) A solution may be illusory that is, it may assume the form -. " Ex. 1. There are two pieces of cloth of a and a yards respectively. The owner sells the same number of yards of each at b and 6 shillings per yard respectively, he then sells the remainder at c and c shillings a yard, and finds that the prices received for both pieces are the same. Required the number of yards first sold. ac a c The number is 7; TT > b b + c o As a particular case, if a and a are 60 and 80; b and b 10 shillings and 9 shillings; and c and c 4 shillings and 3 shillings, the answer assumes the form - . The answer is in this case indeterminate; in other words, the conditions of the problem are satisfied independently of the number of yards first sold; any number will do. It is not, however, a necessary interpretation of the form -, that it may be replaced by any number whatever. Most frequently this form results from the fact that some frac tion is not in its lowest terms. Solutions of this kind fre quently occur in ordinary equations they may be avoided; and we offer an example simply to show the method appli cable to cases in which they cannot be avoided. Ex. 2. Find two numbers such that the sum of their products by a and b respectively is c, and the difference of their squares d. We have ax + by = c f) O T wa nt~i //
- y <*
o / " CIX ,
- -(-$-)-*
f f.^2 __ 7i2 /y>2 9/y/ >/ v i/ 1 * _L 7i2 /7 i 9 I U/ j Jit AiLLJU y T^ U Coy ac ft V - (a 2 - 1 2 ) (c 2 + Wf) a? -IP To find the solution when a = b, we observe that (taking the negative sign of the square root) x = - . This arises from the fact that some power or root of a - b is common to the numerator and denominator of the frac tion. To divide this out, we may put a 2 - b 2 =pa*, and we shall get ac acV 1 -p , . - &c. when a is written for b, and for p. (4.) A solution may be introduced by the operation. In the example last given, the positive sign presents us with a solution introduced by the operation, which, Avhen a = b, is not a solution of the problem at all. f> For in that case the two equations become x + y = - ft jC 2 - y 1 d; the latter of which is at once reduced to the simple equation x - y = by means ol the former. Accord ingly, both equations are in this case simple equations, and can admit of only one solution. (5.) As a solution may be introduced by the operation, so may it be dropped out, even when the operation is a perfectly legitimate one. Taking reciprocals, we have 1 1 V(2a+l) +*/(+ 4) _ V(4a+4)+V(3;e+7) x-3 a-3 or J(2x + 1) + J(x + 4) = V( 4a > + 4 ) + V( 3 * + 7 )> which either added to the original equation, or subtracted 3 from it, produces x = - - . M But x = 3 is a solution of the equation which has been dropped out by the omission of the common denominator x-3. It is not necessary to point out that a solution may appear under the form *j -a or GO . In neither case can the problem be solved arithmetically. SECT. X. EQUATIONS IN GENERAL. 78. Before we proceed to the resolution of cubic and the higher orders of equations, it will be proper to explain some general properties which belong to equations of every degree, and also certain transformations which must frequently be performed upon equations in order to prepare them for solution. In treating of equations in general, we shah 1 suppose all the terms brought to one side, and put equal to; so that an equation of the fourth degree will stand thus :
- 4 +PX 3 + qx 1 + rx + s = ,
where x denotes an unknown quantity, and p, q, r, s, num bers or fractions, either positive or negative. Here tho coefficient of the highest power of x is unity, but had it been any other quantity, that quantity might have been taken away, and the equation reduced to the above form by rules already explained (Sect. VI). The terms being thus arranged, if such a quantity be Definit found as, when substituted for x, will render both sides = 0, of root; and therefore satisfy the equation, that quantity, whether it be positive or negative, or even imaginary, is defined to be a root of the equation. But we have seen that every quadratic equation has always two roots, real or imaginary; we may therefore assume that a similar diversity will take place in all equations of a higher degree; and this assump tion appears to be well founded, by the following proposi tion, which is of great importance in the theory of equa tions. If a root of any equation, as a; 4 +px 5 + qx z + rx + s = 0, be represented by a, the first side of that equation is divi sible by a; -a; For since # 4 +PX 3 + qx 1 * + rx + s - 0, And also Therefore, by subtraction, But any quantity of this form x* - a", where n denotes a whole positive number, is divisible by x - a (Art. 20, Ex. l Hence, since every term contains a factor of this form, the equation may be written under the form (x - a) (x 5 +PX" 2 + q x + r ) = .
i.e., the expression # 4 +/>a; 3 + <p; 2 + r.r + s is divisible by
