Teaching the Derivation of the Quadratic Formula
Long ago I was teaching (I use the word loosely) a class of college students when we somehow got into a discussion of the quadratic formula for the solution of general quadratic equations of the form , I was not surprised that all of the students correctly knew the formula:
I was very surprised that EVERY student in the class was adamant to assert they had never seen a derivation of the formula. The claim was that it was just a formula they had been given, memorized, and learned when and how to use it. The discussion was particularly alarming given that this was a class of college juniors who had completed a significant part of a mathematics major and who were preparing to be secondary school mathematics teachers.
Subsequent discussion led to the admissions that a teacher had probably presented the derivation; it just was not something they had felt they needed to know and remember. The important thing to remember was the formula. Subsequent experience with the quadratic formula had made it their preferred method for solving quadratic equations, even which simple factoring would have worked (e.g. ).
Further, no one expressed confidence with “completing the square” as a way of solving a quadratic equation. That too had faded from awareness as the formula took over the primary role for solving their quadratic equations.
Is this Knowledge Important?
Is it important for students to learn and understand the derivation of the quadratic formula? Is it important for students to learn and understand the technique of completing the square for solving quadratic equations? In particular, are these topics important for prospective mathematics teachers to know and understand? After all, most applications problems, and assessments require only finding the roots of quadratic equations and the formula is a well practiced tool for such uses. Are mathematics teachers the only ones who need to know the derivation of the quadratic formula?
It's in the Common Core
The ultimate authority these days for whether some mathematics topic is important is the Common Core. After all, it has been endorsed in 47 states. We find this
· CCSS.Math.Content.HSA-REI.B.4 Solve quadratic equations in one variable.
- CCSS.Math.Content.HSA-REI.B.4a Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form.
- CCSS.Math.Content.HSA-REI.B.4b Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
An assessment task is described in this way:
In particular, the common core assessment stresses these two standards of mathematical practice:
MP.1 Make sense of problems and persevere in solving them.
MP.5 Use appropriate tools strategically.
Clearly, the Common Core documents can be interpreted as support of the idea that mathematically proficient students should know and understand the derivation of the quadratic formula and they should be proficient with the entire range of tools for exploring quadratic equations.
Before discussing approaches to deriving the quadratic formula, some assumptions are to be made about prior knowledge of the students. For most curriculum approaches to implementing the derivation, the students will have previously developed proficiency with at least the following:
1. Solving linear equations.
2. Factoring of trinomials
3. (x + k)(x + m) = 0 implies x = -k or x = -m
4. Solving quadratics by factoring
5. Taking square roots
6. Completing the square
7. Using technology to graph equations
Deriving the Quadratic Formula by Completing the Square
The internet has many resources -- lesson plans, videos, and presentations. The videos for the derivation of the quadratic formula are of varying quality but most of them are in the tradition of well-organized exposition and highly procedural.
This video has voice explanation of the steps that are being taken. It is somewhat different in the way that the parameter a in the quadratic is handled. That is, it is factored out of the expression rather than just dividing. The steps and are given below:
The algebra is straight forward but unnecessarily complicated. This way of handling the a coefficient has the added advantage of showing the development of the discriminant throughout the presentation. A simplification is the following:
Another video using this approach is:
The Usual Approach.
The usual approach is subtract c or c/a from each side of the equation early in the process. Most of the videos show (and explain) the following steps:
Examples of this usual approach include:
http://www.youtube.com/watch?v=jHoYuHrmIXc http://www.youtube.com/watch?v=xApGFHDj9BM http://www.youtube.com/watch?v=qEjsqcygHWs http://www.youtube.com/watch?v=f7812n-1XYE http://www.youtube.com/watch?v=W5DeCerOqFw (with no voice) http://www.youtube.com/watch?v=ygZHOrkYEK
http://www.youtube.com/watch?v=hmTXf1ufOKA (with no voice)
Here is an interesting video of the derivation of the quadratic formula by a 6-year-old.
Another way of dealing with the coefficient a
In this video, instead of starting by dividing both sides by a, this approach first subtracts c from both sides and them multiplies both sides by a to get a perfect square in the first term. The steps and explanations are
So if the derivation of the quadratic formula is important to learn and understand, what are some options for student centered lessons. I would characterize the videos linked above to be teacher centered and one suspects most attempts and instruction of the quadratic formula are more teacher centered than student centered.
What are some options for such student centered lessons?
What are some motivations for showing the need and value of a general formula?
How can technologies be incorporated into student investigations of the quadratic formula?
An Alternative Quadratic Formula
A New Challenge
Prove that these two forms of the quadratic formula are equivalent, given that a ≠ 0 and c ≠ 0.
Actually, since they were each derived from the general quadratic, we already know the are equivalent, but a direct proof could be insightful.
It might be useful to look at a couple of examples: