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NOTES ON ALGEBRA. 67
Call small gear x and large gear y ; then x + y must be 108,
because 6^ X 16 = 108, which is the number of teeth in both
gears added together. The ratio is 1 to 3; therefore 3x—y,
Thus:
x + y = 108, transposed to
•r-j-3 jr=108.
4x = 108.
.r = 143.
4
jr = 27 teeth for small gear.
The large gear = 27 X 3 = 81 teeth.
Quadratic Equations.
Equations containing one or more unknown quantities in
the second power are called quadratic equations. If the un-
known quantity only exists in the second power the equation
may be brought to the form x2 = a and x = \/ #7~
This square root may be either plus or minus.
If the unknown quantity exists in both the first and the
second power the equation may be brought to the form x2 4- a x
= d, or it may be brought to the form x2 — a x = b.
The coefficient a may be any number. After the equation
is brought to this form, complete the square of the first member
by adding to both members of the equation the square of half
the coefficient a ; this will make the left member of the equa-
tion a complete square.
Example.
A coal bin is to hold six tons of coal. Allow 40 cubic
feet per ton. (It takes 35 to 40 cubic feet to hold a ton of
coal in a bin). Make the width of the bin 6 feet, and the
length equal to the width and the depth added together. How
deep and how long will the bin be?
Depth = x and length = y.
x y = 240, because 6 times 40 equals 240.
6 + x = y, because width + depth = length.
Thus:
6 x (6 + x) = 240.
$x2
+ 36 x = 240.
Dividing by 6 we have : x2 -j- Qx = 40.
Completing the square : x2 -\- <ox -{ 32 = 40 + 32
.
x + 3 = \/ 40 + 9.
Extracting the square root: x + 3 = \/ 49.
.r + 3 = 7.
x = 7 — 3.
x = 4 feet deep.
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