Concept of discriminants

[skeeter]

As I am preparing for an upcoming unit on Quadratic Functions, I am looking for a connection between the concept of discriminants and their relationship to the roots of quadratic equations. I have read through the standards and am trying to find a connection to a specific standard. Would this concept surface through A.REI.4? Is this a connection that students should build in order to progress through quadratic functions in Algebra I to Algebra II? Any feedback would be helpful. This is my first year teaching Algebra as well as teaching the Common Core Standards. So, I am trying to make sense of the content provided in my textbook along with the content described in the Standards and the progression documents. Thank you for your response.

[Anonymous]

I’m thinking about what I gain by knowing that the value of the discriminant gives me information about the number of real roots in a quadratic. My thought is “not enough” to go beyond having the word as part of my students’ vocabulary.

The standards call for being able to solve quadratic equations by a method appropriate to the initial form of the equation AND recognize and write complex solutions. So, how does the discriminant connection help students progress through this standard? I would argue it doesn’t, and may in fact detract from the focus of the standards. First, it replaces “reasoning” with a “rule”. If I have learned that the square root of a negative number is a non-real result, then I should be able to reason that a negative discriminant gives non-real results. I don’t need to learn a new thing in order to learn something I already know. (reminds me of things like the vertical line test.)

Secondly, a focus on zero, one, or two solutions is a little misleading since even if there are no real solutions, we still need to to recognize (Algebra 1?) and write (Algebra 2?) the complex solutions.

[Bill McCallum]

abienek is right that the discriminant is not required explicitly, although A.REI.4 would be the place to put it. Of course, it’s there implicitly in that it’s the thing under the square root sign in the quadratic formula. No harm in pointing out that that thing has a name, I suppose, but I agree it doesn’t seem necessary and could need to excessive rulifying.

[Author]

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