Remember the good old days when students solved 'quadratic-type' equations? Of course, many are still doing this but it is fast becoming a lost art (and some of you may feel it should be!). It is not required in any state math standards or Achieve's, so there's no reason to mention it, right?
Below you will find a 4th-degree (quartic) polynomial equation. The rational root theorem won't help because there are no rational roots. The graphing calculator won't help because there are no real roots! Ok, maybe Mathematica and other Symbolic algebra software could do this, but who exactly programmed this?
Using substitution to rewrite certain 4th degree equations as quadratics (so-called 'biquadratic' equations) used to be covered in some Algebra 2 or advanced classes. Some of you may feel nostalgic about this. However, our challenge today is to solve this by at least TWO 'radically' different methods and then show the solutions are equivalent!
Here's your equation:
x4 + 3x2 + 4 = 0
(a) Explain, without solving, why this equation has no real roots. Should ALL students in Algebra 2 and beyond be able to answer this one?
(b) Solve, by substituting y for x2 and using the quadratic formula. You should eventually arrive at 4 imaginary solutions. This is the way I was taught to solve it, eons ago.
(c) Solve by completing the square and factoring. [Definitely not the first method I would have thought of way back when...]
(d) Show your results are equivalent. This may be annoying! So, which method is easier in your opinion?
(e) Any other method for finding imaginary solutions?
QUICK OPINION POLL
(1) Completing the square (not to mention factoring) is no longer an important topic and should be deemphasized in our curriculum (or omitted).
By the way, is it explicitly mentioned in your state's math standards for Gr 8-12?
(YES NO)
(2) The equation in this post has little relevance to the 21st century and Dave should be ashamed for publishing such trash. Besides, this topic is not included in the Algebra 2 Standards developed by Achieve and ADP.
(YES NO)
You've perhaps assumed that since I've been discussing and complimenting Achieve's standards and the new Algebra 2 End of Course Exam, that I would no longer advocate exposing students to this kind of traditional mechanical 'exercise.'
Well, I taught from the AP Calculus syllabus and I still made time to discuss some ideas and methods that were not 'required'! Further, who exactly will be the ones left on this planet who know how to find imaginary roots for this type of polynomial equation that has no real roots! In case you're wondering, this kind of question has traditionally been taught in Asian countries and still is! (Dave, can you document that? Sure...)
Tuesday, October 2, 2007
But it's not in the Standards: Finding imaginary roots, completing the square, factoring and other 'obsolete' topics...
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20 comments:
Nice post. I too mourn the death of completing the square...
I'm not sure what you mean with part (d), though. The quadratic formula is derived by taking a general quadratic equation and completing the square. So they're always equivalent. What kind of answer do you expect here?
And to weigh in on your opinion poll, from the point of view of a college calculus teacher, algebra (including factoring) is essential in calculus, for instance for finding critical points of functions. Sure, calculus tells you to take the derivative, but how do you solve the equations?
Thanks for the positive comments, Matthew. Some assume that I'm a 'radical' reformer because I write all of these investigations, but they don't understand me at all!
Yes, of course, the quadratic formula, is derived from completing the square so the 'methods' are equivalent. However, direct use of the QF leads to the roots in a very different form from directly completing the square in this case. Showing these forms are equivalent takes some work.
a) yes, they should. Even calculator heavy kids might have a shot.
b) me too, I learned this. Thought it was minor and obvious at the time, but I reacted that way to anyone dragging 5-10 minutes of material into 55 minutes of discussion and practice.
c) amazingly simple! I never would have tried it.
d) just looks like lots of steps. Or maybe I am doing it wrong? But the nice part of c) is not having to run so far with it.
e) none that jump out at me.
1) No. Idk.
2) No.
Hi,
Maybe I am being true to my nom de clavier here, but I do not see
x^4+3x^2+4 having purely imaginary roots. It seems to have complex roots. Maybe you meant x^4+5x^2+4 ?
To solve x^4 + 3x^2 +4, you can complete the square two ways: the first way (x^2+3/2)^2+7/4 =0, which will give you two complex values for x^2, from which you have to find x. Straightforward, but tedious. The second way is to write it as (x^2+2)^2 - X^2 =0, which will give you two quadratics, which can be easily solved.
To solve X^4+5X^2+1=0, I think the best way to do it is by factorization. Students should have this skill and use the quadratic formula only as a backup.
TC
tc--
You are never clueless!
I used the term 'imaginary' only to distinguish them from real roots. I wasn't implying 'pure' imaginary, i.e., of the form 0+bi. 'Complex' would have avoided this confusion. Your 2nd method of completing the square is what I was looking for. It's not something students commonly see these days.
oh. i see.
i completed the square like so:
x^4 + 3x^2 = -4
x^4 + 3x^2 + 9/4 = -4 + 9/4
(x^2 + 3/2)^2 = -16/4 + 9/4
etcetera (and of course got
the same answers -- in the
same form! -- as one obtains
with the quadratic formula [QF]).
better to omit QF itself
than omit "completing the square".
mathematics isn't a bunch of rules
we follow like trained chimpanzees
but a collection of techniques
for understanding the universe.
i finished a math major
without having memorized QF
(i only learned it when i became
an instructor ... it seemed unlikely
that students would take to
"if it doesn't factor,
just complete the square").
it may be worth mentioning that
while QF is *derived* by c.t.s.,
it can "easily" be *checked* without
(just "plug in" on Ax^2+Bx+C
and turn the crank; good exercise).
VME
tc--
I think I might have muddied the waters with 'completing hhe square'. Of course, if you do it hhe traditional way,i.e., adding 9/4 to each side, this will lead to the same result as QF. I should have made it clear that I had intended the +x^2, -x^2 method (but I didn't want to give it away!).
Hi Dave,
Maybe the wording could say: Think of two ways of completing the square. One will lead you to the quadratic formula, but is there another way that might simplify the problem?
Of course, only the second way leads to a factorable form, so you did have a hint there.
TC
Back when I learned the quadratic formula (35 years or so ago), we derived it by completing the square of ax^2+bx+c=0.
And quite a few times since then I've re-derived the quadratic formula that way when the details had become fuzzy through disuse.
Is there another way of deriving the quadratic formula?
Rachel,
I guess you're making a strong case for keeping this procedure! You are not alone...
I think I may have confused several readers by my directions which tc has improved upon considerable. Ther 'completing the square' method i used was a variant of the traditional one:
x^4 + 4x^2 + 4 - x^2 = 0, etc.
Now I'd like to extend that approach a bit:
(1) Use the above method to find the four 4th roots of -4 without trig or polar form, i.e., purely algebraically.
(2) Ok, you solved x^4 + 4 = 0. Now find a general expression for all positive integers k such that the above method can be applied to solve x^4+k=0.
Dave:
Interesting problem. Since I screwed up the last problem let me try this one. x^4 + k = 0. The key here is finding an expression for the sqrt(i). This can be done by equating
(sqrt(i))^2 = i = (a + bi)^2 where a and b are reals. You get two answers
+/-sqrt(2)/2(1 + i). From this it is easy then to write out the results.
Nice job, cecil.
Actually, I probably didn't make the question clear enough, but I'm looking for an expression for those values of k for which one can use that special 'completing the square' technique in which we produce a difference of squares:
k=4: x^4+4 = (x^4+4x^2+4) - 4x^2
k=64: x^4+64 = (x^4+16x^2+64) - 64x^2
What would be the next value of k for which this can be done? Also, find a general formula. This leads to solutions which do not require expressing the square root of i in a+bi form!
Well, no takers on my challenge, no doubt caused by my 'typo' on the 2nd example!
It should have read:
k=64: x^4+64 = (x^4+16x^2+64) - 16x^2
Sorry 'bout that...
Now, have you figured out the next value of k? Ok, I'll give it to you and then you should be able to give the general formula for k:
Next value: k = 324
By the way, while you're at it, why not determine the general solutions for x using this procedure rather than determining the 4th roots of k by other methods. Roots should be in a+bi form of course.
Not clear what you are asking for, Dave.
For any positive k, we can write
x^4+k =0, complete the square, and finally get
x= k^(1/4) (1+i)/sqrt(2) and the other variations thereof by replacing the + with a - and also negating the whole expression.
Are you asking for values where you will get a complex integer for the answer?
TC
Yes, tc! Again, my directions have been far form clear lately (brain freeze, I guess!).
I was looking for conditions on k which produce solutions of the form:
+/-t +/-ti, where t is an integer, when we solve the equation:
x^4 + k = 0, k>0.
This is equivalent to values of k for which we can express x^4 + k as a difference of 2 squares.
Ok, here it is:
Let k = 4t^4.
Then x^4 + 4t^4 =
(x^4+ 4t^2x^2+4t^4) - 4t^2x^2 =
(x^2+2t^2)^2 - 4t^2x^2 =
(x^2+2t^2+2tx)(x^2+2t^2-2tx) =
If we set this equal to zero and solve, we obtain (by QF) precisely those 4 solutions mentioned above, complex integers if you will:
+-t +-ti.
The first few values of k for which this applies are:
t=1: k = 4
t=2: k = 64
t=3: k = 324
t=4: k = 1024
t=5; k = 2500
Time to put this to rest I guess!
Dave, I'm stopping by to say that I appreciate these kinds of posts and ensuing discussions.
Pardon my ignorance, but are there really people seriously saying that completing the square shouldn't be in the curriculum?
Yes, Myrtle, there are some who feel that completing the square has limited application (outside of deriving the quadratic formula) and should be deemphasized. I am not one of those individuals. I will now state the 'obvious': The technique is useful for changing the form of various quadratic-type expressions from equations of conics to integrating rational expressions in calculus. I will explore this further by looking at various states' standards and the algebra benchmarks developed by Achieve and the American Diploma Project. This is not about reform vs. traditionalism. It's about making informed decisions based on the skills and conceptual understandings needed by students who intend to continue their study of mathematics. Then there is the issue of reducing the number of required topics if more and more students nationally are to complete Algebra 2 successfully. This was one of Prof. Steen's points in his interview. Now I'm opening up another can of worms...
There is a discrepancy between the math people need for "life" and the math that people need for further math study. Should we be "streaming" kids for further math study or not by Alg2? I understand the desire to get more kids through the "real life" math skills up through algebra 2.
I can think of two solutions:
1) Offer 2 versions of algebra 2, one of which satisfies the prerequisites for precalculus, and teaches things like completing the square; the other version would be more "life skills" oriented and satisfy the goals of getting more kids up to that level of math without overwhelming them with calculus-bound topics they don't really need
2) Squeeze some of the "further math study" stuff into precalculus and keep Algebra 2 purely "practical". I'm not sure whether more could be squeezed into precalc or not.
California Standard for Algebra I # 14.0) Students solve a quadratic equation by factoring or completing the square.
California Standard for Algebra II #8.0) Students solve and graph quadratic equations by factoring, completing the square, or using the quadratic formula. [And since quadratics with complex roots can't be graphed by factoring, completing the square is at least sometimes necessary.]
Completing the square by adding and subtraction (b/2a)^2 is practically necessary in order to deal with transformations of quadratics, for reducing a jungle of different quadratics to variations of the parent function, for simplifying the process of graphing. Your method for completing the square is new to me, though.
H--
Thank you for saving me the time to look up these standards. I haven't seen that much specificity in some other states' standards, so this is encouraging.
The special completing the square form I used was a standard technique when different advanced methods of factoring used to be taught in an advanced algebra class. Nowadays, it's difference of squares and trinomials and that's about it. The rest is generally considered less important, fairly obsolete techniques that can be handled by a symbolic algebra manipulator. Even the sum and difference of cubes is deemphasized to put it kindly. My take is that for the majority of students, this may be acceptable, but for those pursuing advanced mathematics, some facility with these methods is worthwhile for trig, calculus and beyond.
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