An Equation Which May Be More 'Complex' Than It Appears!

Maybe I should rename this blog to Saturday 'Morning' Post. After all, no one reads that either anymore!

As the school year comes to a close (and I'm assuming it's already over for some), here's an innocent-looking equation which might be worth discussing with your advanced algebra/precalculus students now or next year. I might have considered saving this for our next online math contest but it's complex nature makes it more suitable for discussion in the classroom than on a test. Have you seen exercises like this in your Algebra or Precalculus texts? Do students often delve beneath the surface of these? It's kind of like a black box. We often feel we simply cannot reveal too much of the mystery here or we will not finish required content. Well, you know my philosophy of 'less is more' and I don't even live in Westport, CT. (Ok, that's a post for another day!).

SOLVE (by at least two different methods):

2a^-3/2 - a^-1/2 - a^1/2 = 0

Preliminary Comments/Questions/Issues

• Is the term solve ambiguous here, i.e., should we always specify the domain to be over the reals or over the complex numbers or is that understood in the context of the problems? I'm guessing that most advanced algebra students learn that the domain of the variable or solve instructions may impact on the result, but, that is precisely one of the objectives of this problem.
  
• Should students immediately change all fractional exponents to radical form? OR use the gcf approach (which requires strong skill)?
  
• It's not hard to guess that 1 is a solution but is it the only solution? Can we make a case for -2 being the other solution? The graph doesn't reveal this and surely, -2 doesn't make sense or does it....
  
• Is there ambiguity in raising a negative real number to a fractional exponent (never mind raising i to the i)? Why? Isn't there a principal value for such an expression? How is it defined? This problem raises fundamental and sophisticated issues about numbers which can be taken as far as one chooses to go Just how complex can complex numbers get?
• What is the role of the graphing calculator here? Mathematica? Wolfram Alpha? In addition to verifying solutions or determining answers, can these tools also be useful in clarifying ideas or raising new questions?
  
• Students (and the rest of us) are now capable of quickly filling in the gaps in their knowledge base by visiting Wolfram's MathWorld or Wikipedia for more background. Should this impact on how we present material? Typically, in the pre-web days teachers would avoid opening up a can of worms like complex solutions here, but, with your more capable groups, the sky's the limit now IMO...
  

But it's not in the Standards: Finding imaginary roots, completing the square, factoring and other 'obsolete' topics...

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...)

Investigating an SAT Algebra Problem - Going Beyond the SAT Strategy for Deeper Meaning

[Update as of 6-2-07: Solutions to most parts of the problem are now posted in the Comments.]

The following problem is typical of a somewhat difficult Algebra 2/Precalculus type of SAT problem. Students who have been shown the 'quick and easy way' to solve this definitely have an advantage when taking the test. This has been developed into an activity for Algebra 2 students who might be taking the SAT on June 2nd. However, the activity explores more than just a strategy for solving the problem efficiently to get an answer for the test. Students will be asked to solve the problem the traditional way as well and analyze why this question should not appear on the test. Two separate graphical interpretations are included to deepen student understanding, one for Algebra 2 students and a more sophisticated one for the Precalculus class. Is it worth spending 25 minutes or more on one problem in class? I'll let you judge...

Consider the system of equations:
1/x + 1/y = 1/4
1/(x+y) = 1/3

(a) (SAT-type of thinking): Show that xy = 12 (without solving for x and y).
Note: I'm giving the 'answer' here so that students will focus on the method; also for part (f).

(b) (Algebra 2): Solve the system (i.e., find all possible solutions for x and y). Write your answer(s) as ordered pairs.

(c) Verify, algebraically, that your solutions satisfy the original equations.

(d) Explain why this exact question should not appear on the SATs (although this type of question has appeared frequently).

(e) (Precalculus extension): Graph the system:
1/x + 1/y = 1/4
1/(x+y) = 1/3
Notes/Hints:
You must graph each equation separately. Do not replace the first equation by xy = 12.
Hint: Solve each equation for y.

(i) Explain clearly why the graph of the first equation has both vertical and horizontal asymptotes. Label these in your graph.
Note: The customary way is to solve for y first. See if you can also determine the asymptotes without changing the form of the equation!

(ii) Does your graph of the first equation contain the origin? Explain why or why not.

(iii) How many solutions of the system are evident from the graph? Explain the reason for this.

(f) (Algebra 2): Now solve the system
x+y = 3
xy = 12

(i) In what way is this system equivalent to the original system?
(ii) How many solutions of this system are evident from the graph? Explain the reason for this.