Page 307 - Theory and Problems of BEGINNING CHEMISTRY

296 SCIENTIFIC CALCULATIONS [APP.

The acceleration is 3.00 mi/h per second. This is an example of one of the few times when two different units are
used for the same quantity (time) in one value (the acceleration).

QUADRATIC EQUATIONS
A quadratic equation is an equation of the form
2
ax + bx + c = 0
Two solutions are given by the equation
√
2
−b ± b − 4ac
x =
2a
This equation giving the values of x is known as the quadratic formula. Two answers are given by this equa-
tion (depending on whether the plus or minus sign is used), but often, only one of them has any physical
significance.

EXAMPLE A.9. Determine the values of a, b, and c in each of the following equations (after it is put in the form
2
ax + bx + c = 0). Then calculate two values for x in each case.
2
2
(a) x − x − 12 = 0 (b) x + 3x = 10
Ans. (a) Here a = 1, b =−1, and c =−12.

2
−(−1) ± (−1) − 4(1)(−12)
x =
2(1)
Using the plus sign before the square root yields
√
+1 + 1 + 48
x = = 4
2
Using the minus sign before the square root yields
√
1 − 49
x = =−3
2
The two values for x are 4 and −3. Check:
2
(4) − (4) − 12 = 0
2
(−3) − (−3) − 12 = 0
(b) First, rearrange the equation into the form
2
ax + bx + c = 0
In this case, subtracting 10 from each side yields
2
x + 3x − 10 = 0
Thus, a = 1, b = 3, and c =−10. The two values of x are
√ √
−3 + 9 + 40 −3 + 49
x = = = 2
2 2
√
−3 − 49
and x = =−5
2

Conversion to Integral Ratios
It is sometimes necessary to convert a ratio of decimal fraction numbers to integral ratios (Chaps. 3 and 8.)
(Note that you cannot round a number more than about 1%.) The steps necessary to perform this operation follow,
with an example given at the right.