56 
2. Sometimes it is convenient to express by 
letters, not the unknown quantities themselves, 
but some other quantities connected with them, 
as their sum, difference, &c. from which they 
may be easily derived. 
3. in the operation, also, circumstances will 
suggest a more easy road than that' pointed out 
by the general rules. Two of the original equa- 
tions may be added together, or may be sub- 
tracted; sometimes they must be previously mul- 
tiplied by some quantity, to render such addi- 
tion or subtraction effectual, in exterminating 
one of the unknown quantities, or otherwise 
promoting the solution. Substitutions may be 
made of the values of quantities, in place of 
quantities themselves; and various other such 
contrivances may be used, which will render the 
solution much less complicated. 
General Solution of Problems. 
In the solutions of the questions in the pre- 
ceding part, the given quantities (being num- 
bers) disappear in the last conclusion, so that no 
general rules for like cases can he deduced from 
them. But if letters are used to denote the 
known quantities, as well as the unknown, a 
general solution may be obtained, because, dur- 
ing the whole course of the operation, they re- 
tain their original form. Hence also the con- 
nection of the quantities will appear in such a 
manner as to discover the necessary limitations 
of the data, when there are any, which is neces- 
sary to the perfect solution of a problem. 
Ex. 1. To find two numbers, of which the sum 
and difference are given. 
Let s be the sum given, and d the given dif- 
ference. Also, let x and y be the two numbers 
sought. 
x -j- y zz: s 
X — y d 
d -j- y — s — y 
2y q: s — d 
i — d 
-f -d 
h 
thus, let the given sum be 100, and the dif- 
ference 24. 
, 124 
Then *== Hr 
- d __ . 76 
sT~ — 2 
Equations are either pure or adfected. 
Def. 1. A pure equation is that in which only 
wue power of the unknown quantity is found. 
2. An adfected equation is that in which dif- 
ferent powers of the unknown quantity are 
found in the several terms. Thus, 
a 2 -j- ax 1 = P y ax 2 — b 2 =s m 2 -j- x 2 , are pure 
equations. 
x~ — ax = b 2 , x 2 -{- x 2 = 17, are adfected equa- 
tions. 
Solution of Pure Equations. 
Rule. Make the power of the unknown 
quantity to stand alone by the Allies formerly 
given, and then extract the root of the same de- 
nomination out of both sides, which will give 
the value of the unknown quantity. 
And x — 
and 
= ( : 
:) 62 
=) 38. 
EXAMPLES. 
If a 2 -j- ax 2 — P 
ax 1 = P — a 2 
„ P — a 2 
ax m — b — x - 
ax "> — X”‘ = b - 
b - 
IP — a 2 m t b — c 
— V ~~~a ^=\/ a — 
AtG'fifi R a; . 
Solution of Adfected Quadratic Equa- 
tions. 
An adfected quadratic equation (corhmonly 
called a quadratic) involves the unknown quan- 
tity itself, and also its square : 
Rule 1. Transpose all the terms involving 
the unknown quantity to one side, and the 
known terms to the other ; and so that the term 
containing the square of the unknown quantity 
may be positive. 
2. If the square of the unknown quantity is 
multiplied by any co-efficient, all the terms of 
the equation are to be divided by it, so that the 
co-efficient of the square of the unknown quan- 
tity may be 1. 
3. Add to both sides the square of half the 
co-efficient of the unknown quantity, and the 
side of the equation involving the unknown 
quantity will be a complete square. 
4. Extract the square root from both sides of 
the equation, and by transposing the above- 
mentioned half co-efficient, a value of the un- 
known quantity is obtained in known terms. 
The reason of this rule is manifest from the 
composition of the square of a binomial, for it 
consists of the squares of the tw r o parts, and 
twice the product of the two parts. 
The different forms of quadratic equations, 
expressed in general terms, being reduced by 
the first and second uarts. of the rule, are these s 
1. x 2 -j- = P 
2. x 2 — a-x xz. P 
3. x 2 — ax — — P 
Case 1. 
Case 2. 
: -|- a x =?. P 
! 4- ax + ~P 4 - — 
l 1 4 '4 
, a . / , . a 2 
+ 2“ = ± S/ + 4~ 
. / , , a 2 a 
“ ± V 6 + T“ T 
ax — P 
a 2 , c 2 
: ± \A 
+ T 
Case 3. x 2 
= f±xA‘ + f 
■ — ax zit — P 
, . a 2 a 2 
— a 2 A — b‘ 
1 4 4 
* - V = ±v/ X 
p 
x — — ± v /— - P. 
% 4 
L Every quadratic equation will have two 
roots, except such of the third form whose roots 
become impossible. 
2. In the two first forms, one of the roots 
must be positive, and the other negative. 
a 2 
3. In the third form, if — ■, or the square of 
4 
half of the co-efficient of the unknown quan- 
tity, be greater than P, the known quantity, 
a 2 
the two roots will be positive. If — be equal 
' . a 2 
to P, the two roots become equal ; but if — 
is less than P, the quantity under the radical 
sign becomes negative, and the two roots are 
impossible. 
4. If the equation express the relation of mag- 
nitudes abstractly considered, where a contra- 
riety cannot be supposed to take place, the ne- 
gative roots cannot be of use, or rather there 
are no such roots ; for 'the" a negative quantity 
by itsiif is Unintelligible, and therefore tk# 
square root of a positive quantity must he posn 
tive only. 
Solution of Questions producing Qua- 
dratic Equations. 
The expression of the conditions of the ques- 
tion by equations, or the stating of it, and the 
reduction likewise of these equations, till we 
arrive at a quadratic equation, involving only 
one unknown quantity and its square, are ef- 
fected by the same rules which were given for. 
the solution of simple equations. 
Ex. 1. One lays out a certain sum of money 
in goods, which he sold again for 24/. and gained 
as much per cent, as the goods cost him: I de- 
mand what they cost him ? 
If the rhoney laid out be y 
The gain will be 24 — y 
But this gain is \ 2400 — lOOy 
(y t 24 — y\\ 100) y 
Therefore by ) 
question £ 
J 
And by mult, and tr.y 2 -j- TOOy = 2400 
Completing the ? y 2 -j- lOOy -|- 50 > ) 2 — 2400 
square £ -f- 2500 = 4900 
Extract the root y 50 — Ar \/ 4900 — 70 
Trans. - y — -j- 70 — 50 = 20, or 
— 120 . 
The answer is 20/. which succeeds. The other 
root, — J 20, has no place in this example, a ne- 
gative number being here unintelligible. 
To find two numbers whose sum is 100, and, 
whose product is 20.59. 
Ex. 2. Let the given sum 100 — a, the pro- 
duct 2059 — b, and let one of the numbers 
sought be x, the other will be a — x. Their 
product is ax — x 1 . 
Therefore ax — x 2 — b, or x 1 — ax — — b 
per cent. 
2400— lOOy 
Compl. the sq. x 2 
ax h — — 
* 4 4 
Ext. sj 
Transp. 
And the other 
number 
— .v - -- = + v / — - 
2 “ V 4 
J < - t 
V 4 
} a fa 1 
\ a -*—2 + v T 
b. 
By inserting numbers, x — 71 or 29, and 
a — x =29 or 71, so that the two numbers 
Sought are 71 and 29. 
Here it is to be observed, that b must not be 
a 2 
greater than , else the roots of the equation 
would be impossible; that is, the given product 
must not be greater than the square of half the 
given sum of the numbers sought. This limita- 
tion can easily be shewn from other principles ; 
for the greatest possible product of two parts 
into which any-n umber may be divided, is, when 
each of them is a half of it. If b be equal to 
— there is only one solution, and x = — , 
4 2 
a 
also a — x = 
Of Indeterminate Problems. 
It may be observed, that if there are more 
unknown quantities in a question than equa- 
tions, by -which their relations are expressed, 
it is indetermined. In other circumstances, such 
problems are resolved by Various methods, not 
to be comprehended in general rules. 
Ex. 1. ’To divide a given square number into 
two parts, each of which shall be a square num- 
ber. 
There are two quantities sought in this ques- 
