TÖ THE RESOLUTION OF PROBLEMS, 
99 
it will become 
3 — 6 X 24 — 38 — 2 — 3 X 36 — 83 
2—4X 3— 6—2 — 3 X 3 — 10 
= 3; whence, also, we find 
bd — ne 2 — 3 
■) = 
2 — 3 
12 — 4 — 5 
Having exhibited a variety of examplesof the use and 
application of Algebra, in the resolution of problems 
producing simple equations, I shall now proceed to give 
some instances thereof in such as rise to quadratic equa® 
tions; but, first of all, it will be necessary to premise 
something, in general, with regard to these kinds of 
equations. 
It has been already observed, that quadratic equations 
are such wherein the highest power of the unknown 
quantity rises to two dimensions; of which there are 
two sorts, viz. simple quadratics, and adfected ones* 
A simple quadratic equation is that wherein the square 
¡only of the Unknown quantity is concerned, as xx —ab m „ 
but an adfected one is, when both the square and its 
root are found involved in different terms of the same 
equation, as in the equation x 2 q- 2ax — bb. The re 
solution of the first of these is performed by, barely, 
extracting the square root, on both sides thereof: thus 
in the equation x 2 — ah, the value of x is given — \/ah 
(for if two quantities be equal, their square roots must 
necessarily be equal). The method of solution when 
the equation is adfected, is likewise by extracting the 
square root; but, first of all, so much is to be added to 
both sides thereof as to make that where the unknown 
quantity is a perfect square! this is usually called com 
pleting the square, and is always done by taking half 
the coefficient of the single power of the unknown 
quantity, in the second term, and squaring it, and 
then adding that square fa both sides of the equation. 
Thus, in the equation xx + 2ax — bb, the coefficient 
of a; in the second term being 2a, its half will be a, 
which, squared and added to both sides, gives + sax 
h 2