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THE APPLICATION OF ALGEBRA 
gives bx 4- ba — hx — ax' 4 a'x, that is, ba — ax* 4- 
a'x; whence, dividing by a, and completing the square, 
we have x 2 -f ax + \a 2 — b + 4a 2 ; therefore x 4 \a — 
\/b 4- 4a 2 , and consequently x — \/ bf ^a l — \a — 
SO, the lesser number, whence x 4- a — 30, the greater 
number* 
IHOCLEM LVII 
To find two numbers whose sum is 80 fa J, and if they 
be didded alternately by each other, the sum of the quo 
tients shall be 34 (bJ. 
If one of the numbers be x, the other will be a — r, 
x cl - x 
and we shall therefore have— \ - b: which 
a — x x 
equation, brought out of fractions, becomes x 2 4 a 2 — 
Sax 4- x 2 rr abx — bx'; and this, by transposition, 
gives 2x 2 -f bx* — 2«x — abx — — a 1 , that is; 
2 4-6 X x 2 — 2 4- 6 x ax ——a 2 ; whereof both sides 
being divided by 2 4- b, we have x 2 — ax — — 
two, are the numbers that were to be found. 
PROBLEM LVII 1* 
To divide the number 134 (a) into three such parts,- 
that once thefirst, twice the second, and three times the 
third, added together,may be — 278 (b J,and that the sum 
of the squares of all the three parts may be — 6036 (cj. 
Let the three parts be denoted by x, y, and 2, respec 
tively; then, from the conditions of the problems, we 
shall have these three equations. 
x 4- y 4- z = a, 
x 4- 2y 4-32 =6, 
x* 4- y 1 4- a* ~ c. 
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