College Math Teaching

March 29, 2013

The Quadratic Formula: case study in misunderstanding its meaning (and a moral)

Filed under: basic algebra, editorial, elementary mathematics, mathematical ability — collegemathteaching @ 8:29 pm

I admit that I never dreamed that something as innocent as this picture (a friend tagged me on Facebook) would lead to a sort-of heated argument.

qf

Of course this is the famous quadratic formula; it gives the roots to the following equation: ax^2+bx+c = 0 with a, b, c complex numbers and \sqrt{w} interpreted as the principle solution to (\sqrt{w})^2 = w . In fact this works in any field in which the square root is defined.
This formula is just a trivial consequence of completing the square: assume that a \ne 0 then
a (x^2 + \frac{b}{a}  +\frac{c}{a}) = 0 which implies a (x^2 + \frac{b}{a} + \frac{b^2}{4a^2} +\frac{c}{a}-\frac{b^2}{4a^2}) = 0 which implies (x+\frac{b}{2a})^2 = \frac{b^2}{4a^2}-\frac{c}{a} which implies x + \frac{b}{2a} = \pm \sqrt{\frac{b^2}{4a^2}-\frac{c}{a}} which implies x = -\frac{b}{2a} \pm \sqrt{\frac{b^2 - 4ac}{4a^2}} which is the formula.

But one of my friend’s “Facebook Friends” said:

I’ve never actually used the quadratic equation, i just relied on algebraic reasoning

That is a curious statement; my friend remarked that the quadratic formula WAS “algebraic reasoning.” I was curious as to what the comment meant so I posted

Ok, solve 2x^2 + 13x – 17 = 0 without using the quadratic formula OR completing the square (the two are actually the same thing).”

Then came the following response:

when you say 2x^2, do you mean (2x)^2? if so then it is one. it really is easy. you are talking to someone who took ap classes from an MIT grad without using a calculator. and jason, the quadratic formula is an example of algebraic reasoning, which is to say that there are other methods. I was not trying to imply that the quadratic formula is different from algebraic reasoning. math is the language of logic, so i usually relied on my own means to find the answer, although my means can be convoluted at times.

Evidently this individual didn’t understand the significance of my response. This is evident later:

“You really should reserve that for yourself. If it is a quadratic equation, it will almost always have more than one answer, which is outlined by the +/- part of the quadratic equation. Is 1 a possible answer? yes. Is it the only answer? no. I am referring to the the equation i provided btw. Did I assert that 1 was the ONLY answer? no.”
[…]
“I know, that is why i said it wasnt the one I was referring to the whole time. Why else would I ask to clarify? I already know that if I were to go with 2x^2, it would produce an answer with a decimal. That is because if you follow the quadratic equation you will notice that the number in the square root is 305, and the root of the 305 is pretty messy.”

See where the confusion is? Evidently he (yes, the friend and his Facebook friend is a male) did not understand that, while the quadratic formula (or the completing the square process) yields ALL possible solutions for every true quadratic (a \ne 0 ) that in no way means that one can’t, at times, guess a solution or, at times, find an easy factorization. So if you want to solve the general quadratic and find all solutions, you need this formula or the completing the square process.

Of course, in the complex coefficients case, the answer is frequently ugly.

Side notes: there is a formula for the solution to a cubic (very messy) and for the degree 4 polynomial. However, it is impossible to find a general formula to solve the degree 5 polynomial; this is a reason to learn some Galois Theory from abstract algebra!

The other fields: in general the quadratic cannot be solved if the field is, say, an integer of odd prime order, unless one extends the field by adjoining \sqrt{p-1} where p is the prime in question. This is a good reason to learn some number theory.

Moral
Often, students will put the time and effort into understanding a concept if they know WHY it is important. However, they don’t always appreciate what a formula like the quadratic does. One doesn’t always have to use it, but it
1. Provides a method of obtaining ALL solutions that is guaranteed to work in every case (where a \ne 0 )
2. Proves that, in fact, the solutions always exist and what kind they are (real or complex).

These points are not obvious to every beginner, even some who consider themselves to be “bright” and talented. Such self perceptions are the topic of a different post.

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