For given integer values of A,B,and C, what is the probability that B^2-4AC
is a perfect square. (We can stick with integers here; if you want to
include decimals to one or more places, then it's just a matter of
increasing the upper limits of A,B,and C.)
Well, even though I should be preparing to get away for vacation, this is
the kind of problem that is irresistible to me. So I wrote a program to
look at all the triplets of integers A,B,and C from 1 to a chosen limit. It
runs on my TI-83 and I can send it to anyone who is interested. The program
asks for an upper limit of A,B,and C and then tests whether B^2-4AC is
positive and a perfect square. My program doesn't look at negative values
of A,B,and C; perhaps yours or your students' will :-) . It does however,
include triplets that we might consider duplications, like 2,4,2. But my
original premise that nature doesn't give us (many) factorable trinomials
requires that all factorable combinations be counted. One byproduct of the
program is that it prints out factorable triplets, but of course I don't
condone its use for making up factoring tests :-) .
Here are the results:
A,B,C from 1 to 10: probability 0.07 (74 out of 1000)
A,B,C from 1 to 20: probability 0.05 (394 out of 8000)
A,B,C from 1 to 40: probability 0.03 (2016 out of 64000 -over an hour on
the plucky little box)
Do we see a limit here?
So I've done what I was told and learned how to factor any trinomial, even
36x^2+39x+10, and now I'm the dept specialist to whom all factoring will
come. Too bad that I'm going to fail to produce a result almost all the
time. Also too bad that we've just changed to metric and g=9.8 m/sec/sec
instead of good old 32 ft/sec/sec. (I know this discussion doesn't respond
to the recent posts that factoring sharpens one's mind and makes one think;
but those arguments remind me of the 19th century defense of Latin and Greek
requirements. There are lots of things that can sharpen minds, such as
writing programs or estimating roots of randomly chosen polynomial functions.)
And finally, as I sign off for vacation, please don't take any of this
personally. I know that the primary reason that factoring is taught is that
it is in the textbooks and in some of the placement tests we use, not
because professors deeply believe in it.
-Sandy
_______________________________
William J. (Sandy) Wagner
127 O'Connor St., Menlo Park, CA 94025
(650)328-8657 (voice) (650)323-1035 (fax)
sandyw@best.com
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