Resolution of Algebraic Equations. 283 
dratic, only reverted; for, the quadratic surd it contains is 
frequently incapable of further reduction. Therefore, gene- 
rally speaking, the degree of the equation is not altered ; only 
the place of the index, which being first affixed to the un- 
known quantity, is now transferred to the known ones. But 
nevertheless, this resolution is, in all cases, equally true and 
direct ; for, involving no other radical than belongs to the de- 
gree it relates to, it faithfully exhibits the nature of the roots* 
and is always rational or real, or not, according as they are so. 
18. If, instead of seeking, a priori, .the formula of resolution, 
we attempt to find the roots simply, we may instantly trace a 
constant connection between them, or at least between their 
differences; which (however the quantities are varied) are 
always related in the same manner, being [a — b) and ( — a -f- b) 
the same quantity with different signs, and consequently their 
squares precisely the same. From which it appears, that the 
equation of those differences will always want the second term 
or be a pure quadratic ; and that of their squares will be a per- 
fect binomial square, having both roots equal ; which roots may 
therefore, by the reasoning in Art. 1 1, be certainly found. But 
the inference is just the same as before : the equation is not 
lowered in degree ; the equal relation is brought no nearer than 
between the squares of the differences ; and, when they are found, 
the same quadratic surd must be used to arrive at the roots 
themselves. This formula of resolution (,r = ^— 1 — — 12 J is 
the same given for quadratics in every algebra ; but it is not 
usually remarked, or perhaps understood, that the whole opera- 
tion, however varied in appearance by setting about to com- 
plete the square (as it is called) or to destroy the second term. 
