Silly applications of factoring quadratic polynomials

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ouroborosOne of the nicer talks that I attended at the Joint Mathematics Meetings in DC was given by Jeff Suzuki, on “A History of College Algebra in the United States During the Nineteenth Century“.   Suzuki’s talk focused on equation-solving, and he noted that factoring hadn’t been a significant equation-solving tool throughout that era.  Equations were solved by other means (e.g., for quadratics:  completing the square, or the quadratic formula;  for higher order equations, roots might be found/estimated by bisection or other iterative approximation schemes).

This reminded me of one of my pet peeves from teaching algebra.  Modern elementary algebra texts teach four ways of solving quadratic equations:  graphing (and finding x-intercepts by inspection), factoring, completing the square, and the quadratic formula.  Most current books aimed at the “college algebra” market emphasize factoring.  (One of my daily reads, jd2718, has an interesting take on the role of factoring in HS algebra.)

What irked me when last I taught college algebra were the “applications” of quadratic equations.  (Too) Many of the applications amounted to numerical exercises:  “Find two numbers whose product is 45 and whose difference is 4″ was a typical example.

The intent is that a student will introduce one or two variables, representing the two numbers as either x and x+4 or as x and y, and eventually arrive at the equation x^2+4x=45, either directly or via a system of equations.

From here, we solve the related equation x^2 + 4x - 45 = 0, by finding two numbers whose product is -45, and which add to +4:  that is, two positive integers whose product is 45 and whose difference is 4.

The usual technique for achieving this is to list all integer products which equal 45, and find the factorization by inspection.  But of course this amounts to a direct solution of the original word problem, without recourse to any algebra whatsoever.

Granted, there are some advantages to this algebraic method, which become clear when you consider situations with non-integer solutions.  For example, if we want to find two numbers whose sum is 6 and product is 10, trying a brute force attack is unlikely to work.  Setting up the problem algebraically, and reducing it to solving x^2 -6x + 10 = 0, allows us to apply another paradigm — completing the square — to eventually find the two numbers 3+i and 3 - i.

I suppose that there just might be some slight pedagogical value in having students see that solving their quadratic equation is equivalent — literally and explicitly — to solving the original word problem.  But this amount of circularity always struck me as a bit daft, and I feared the day when an eye-rolling student in the back of the class was going to point out that the Emperor was underdressed.

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5 Responses to “Silly applications of factoring quadratic polynomials”

  1. jd2718 Says:

    Translating the ugly descriptions in words into equations is a completely separate skill, a bit old-fashioned, that we still teach in pre-post.

    And once we start teaching it, we reapply it to every new type of equation we encounter. It would be better not to bother with applications.

    Actually, one of the nicer applications I’ve encountered was an extended problem involving profit… We had cost per item + fixed cost, and as we raised the price the number of units sold fell… it came from an awful curriculum that I helped get thrown out of my district. (See this

    I still maintain that factoring per se has value and should be taught, but I don’t know how much value that translation skill has.

    Jonathan

  2. mbork Says:

    In Poland, the *only* method we teach to do quadratics is the quadratic formula. If a student is supposed to factorize a quadratic, he first finds the roots (using the formula) and then writes down the factorized form.

    Also there *is* a problem with really everyday, not artificial, problems… A classical one is to maximize/minimize something using the formula for the vertex of a parabola – not really useful in everyday life, but somewhat closer;).

  3. TwoPi Says:

    mbork: Very interesting. In the US curriculum, completing the square is introduced first, and used as a means of deriving the quadratic formula and the formula for the vertex of a parabola. How are those results derived in your approach?

    When I was a student, completing the square was used quite a bit in analyzing quadratic equations in two variables (conic sections), but so far as I can tell all of that analytic geometry content has fallen out of our precalculus and calculus curricula.

    Completing the square is still present in the integral calculus…. For example, in finding \int \frac{1}{\sqrt{ x^2 + 4x + 5}} \; dx, one typically starts by completing the square under the radical before either doing a trigonometric substitution or applying a table of integrals.

  4. TwoPi Says:

    Perhaps instead of factoring quadratics, we should be solving graph theory puzzles!

    (Isn’t math the coolest? That post at The Number Warrior is fantastic.)

  5. mbork Says:

    Well, when I was at school, the quadratic formula was derived by completing the square; I can’t remember whether and how the formula for the parabola vertex was derived. Nowadays, I’m afraid, the formulae are just given to the students without justification-but I’m not sure…

    In fact, math syllabi in Poland tend to contain less things (and they are undergoing heavy changes these times), but the teachers are supposed to go in more depth – for instance, to solve atypical problems and to teach creative thinking. It is yet to be seen whether this approach will work.

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