ALG LBRA . 
live square root of 5 x -|- 10. The reason is, 
that 5 x -J- 10 is the square of — /y/5r-j-10 
as well as of -j-\/ 5x -j- 10; thns by squar- 
ing both sides of the equation 5 x -)- 1 0 
= 8 — x, a new condition is introduced, and 
a new value of the unknown quantity corres- 
ponding to it, which had no place before. 
Here, 18 is the value which corresponds to 
the supposition that x — ,y/5 a 10 = 8. 
Every equation, where the unknown quan- 
tity is found in two terms, and its index in 
one is twice as great as in the other, may be 
resolved in the same manner. 
Ex. 4. Let is -f- 4 z? = 21 
« — 4 a 2 -j— 4 = 21 — 4 = 2 5 
2 =± 5 
^ = ±5—2 = 3, or — 7. 
therefore z = 9, or 49. 
by extracting the sq. roots, x +y = ± ii 
and x — u — ■ 4-3 
therefore, 2 x — ^ 14 
x = 7, or — 7 
and y — 4, or — 4. 
PROBLEMS PRODUCING QUADRATIC 
EQUATIONS. 
Prob. 1 . To divide a line of 20 inches into 
two such parts, that the rectangle under the 
whole and one part, may be equal to the 
square of the "other. 
Let x be the greater part, then will 20 — x 
be the less, 
and x 2 - = 20 — x . 20 = 400 — 20 * by the 
question. 
x 2 20 x — 400 
20 x + 100 = 400 -j- 100 = 500 
a'4-10=± v /500 
x = + V'bOO — 10, or— ^500 — 10. 
Ex. 5. Let y 4 — 6 y 2 — 27 =0. 
y 4 — 6 y 2 — 27 
y 4 — 6 y 2 -f 9 = 27 -j- 9 = 36 
y 2 — 3 = ± 6 
y 2 = 3 ± 6 = 9, or — 3 
V — ± 3, or ± \/— 3. 
Ex. 6. Let y 6 -j- r xf -f- J- = 0, 
27 ' 
y 6 + r>/= — 
27 
y 6 + ry 3 +r = r 
J 1 J 1 L L 27 
1 / + »—±°v / 1 — y 7 - 
2 :tV 4 27’ 
When there are more equations and un- 
known quantities than one, a single equation, 
involving only one of the unknown quantities, 
may sometimes be obtained by the rules laid 
down for the solution of simple equations ; and 
one of the unknown quantities being discover- 
ed, the others maybe obtained by substituting 
its value in the preceding equations. 
Ex 7. Let \ ^ 65 X To find * and y. 
{ xy=: 28 ) 
From the second equation, 2 x y = 56 
and adding this to the first, x 2 -\-2xy-^~y 2 ~\2\ 
subtract, it from the same, x 2 — 2xy-j-y 2 =:9 
Prob. 2. To find two numbers, whose sum, 
product, and the sum of whose squares, are 
equal to each other. 
Let x -j- y and x — y be the numbers ; 
their sum is 2 x 
their product x 2 — y 2 
the sum of their sqs. 2 x 2 -j- 2 y 2 
and by the question 2 x = 2 x 2 2 y 1 
or x = x 2 -|- y 2 
also, 2 x = x 2 — y z 
therefore, 3 x — 2 x 2 
5 
2 x = x 2 — y 2 
or3 = r 
9 „ 9—12 —3 
y = ± 
v'— 3 
x-\-y 
_ 3 + 4/ — 3 
x — y 
3— v/— 3 
■ ~2 
Since the square of every quantity is posi- 
tive, a negative quantity has no square root; 
the conclusion therefore shews that there are 
no such numbers as the question supposes. 
See Binomial Theorem; Equations, nature 
of; Series, Surds, &c. &c. 
Algeera, application of to geometry. 
The first and principal applications of alge- 
bra were to arithmetical questions and 
computations, as being the first and most 
useful science in all the concerns of human 
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