Recently (i.e., in Chapter 17), we have encountered
problems where it is necessary to solve quadratic equations.
Solving these problems used to be a straightforward, but
tedious process. In years gone by it was an opportunity to
demonstrate your algebraic skills and to develop
judgement
in choosing the physically meaningful answer from among the two
roots. Technology (i.e., more powerful calculators) can
relieve the tedium. However, it can't replace your
judgement, and so I have placed some constraints on the
use of calculators in the General Chemistry course(s). In
Chapter 18, we will learn how
to make approximations that lead to a simplification of the
algebra necessary to determine the answer to some problems.
In the "old days", we made these approximations in order
to save time and effort and to reduce the chance of making
a trivial algebraic or numerical error. In fact, some
of the problems that we will encounter simply cannot be
solved exactly, and thus required the use of
an approximation! Now
advanced scientific/graphing calculators make it possible to
solve numerically such problems
with little effort and no more risk of error than that of
any arithmetical computation. That's good in some ways and
not so good in other ways. Well, enough philosophy ... students want to know:
I have established a policy with respect to this, and you
must be aware of this policy and adhere to it in order to
receive credit for your answers. My intent is twofold:
I want to maintain fairness for owners of standard calculators,
who will not have the opportunity to avoid the quadratic solutions
or the approximations.
I want to motivate even the owners of advanced calculators to
use the approximations because the experience of considering
magnitude, relative error, and desired precision is valuable.
So, here is the policy, for all types of calculator owners:
A cell phone CANNOT be used as a calculator on quizzes and/or exams in this course.
If you own a standard calculator, and you make an approximation,
you must verify that the approximation is valid once you
obtain your answer. The method of verification is clearly
described in the lecture notes.
Quadratic equations can be solved exactly, by hand. If you own
a standard calculator, and you do not make an approximation,
then generally, then you'll get two solutions because
we are dealing with a quadratic equation. You must solve
for both solutions, discard the non-physical one, and state
how you knew that it was the non-physical root (e.g., concentrations
cannot be negative).
For polynomials of higher order, you'll have to make the approximation,
so the previous rule applies (i.e., verify that the approximation
is valid).
If you own an advanced/graphing calculator, BUT you decide to
make the approximation and/or solve without an
approximation (i.e., quadratically, without using the "solve"
feature of your calculator) then the same requirements as listed
above apply to you.
Finally, if you own an advanced/graphing calculator, and you decide
to employ the built-in "solve" feature, so that you do not need
to approximate or, in the case of quadratics, use the quadratic formula, then you must
get all of the roots from your calculator, you must show all
of them
on your paper, you must discard the non-physical one(s), and you must state
how you knew which one was the physical root.
Let me summarize:
If you make an approximation, you get one answer, but you must
verify that the approximation is valid.
If you use any other method, you get two or more roots (depending
upon the order of the polynomial) and you must report all roots,
discard all but one root, and provide justification for doing so.
My personal advice:
Make the approximation whenever possible. It's easy, it's usually valid, and
you will finish the course with a better understanding of
physical magnitudes and estimation.