100 THE APPLICATION OF ALGED^A 
f a 2 — b 1 4- a z ; whereof the former part is, now, a 
perfect square. The square being thus completed, its 
root is next to be extracted; in order to which, it is to 
be observed, that the root, on the left-hand side, where 
the unknown quantity stands, is composed of two 
terms, or members; Whereof the former is always the 
square root of the first term of the equation, and the 
latter the half of the coefficient of the second term : 
thus, in the equation, x z -f 2ax f a z = b % 4- a*, before 
us, the square root of the left-hand side, x z 4- 2ax 4- a 1 , 
will be expressed by .r + a (for x 4 a / x 4 a — x z 4 
2ax 1- a 1 ). Hence it is manifest that X + a — 
VP 4- a z , and therefore x — Vb 1 4- a~ — a; from 
which x is known. These kinds of equations, it is also 
to be observed, are commonly divided into three forms, 
according to the different variations of the signs: thus 
x z 4- 2ax — ¿r, is called an equation of the first form; 
x 2 — 2ax — b 2 , one of the second form; and x z — 2ax 
— —b z one of thq third form; but the method of 
extracting the root, or finding the value of .r, is the 
same in all three, except that, in the last of them, the 
root of the known part, on the right-hand side, is to 
be expressed with the double sign Hh before it, x having 
two different affirmative values in this case. The reason 
of which, as well as of what has been said in general, 
in relation to these kinds of equations, will plainly ap 
pear, by considering, that any square, as x z — 2ax + 
raised from a binomial root, x — a (or a—x) is 
composed of three members; whereof the first is the 
square of the first term of the root; the second, a rect 
angle of the lirst into twice the second; and the third, 
the square of the second: from whence it is manifest, 
that, if the first and second terms of the square be given 
or expressed, not only the remaining term, but the root 
itself, will be found by the method above delivered. 
Hut now, as to the ambiguity, taken notice of in the 
third form, where x 2 — 2ax = — ¿>% or x 2 — 2ax 4 
a z — a 2 — b 2 ; the square root of the left-hand side 
may be either x — a, or a>— x (for either of these, 
squared, produce the same quantity) therefore'in the 
former case, x a 4- Va* — 6% and in the latter, 
fc zz a — \/cl~—b 2 \ both which values answer the