The derivation of the quadratic formula is accomplished by solving the standard form of a quadratic equation by completing the square. As mentioned earlier, I tell students that an extra credit question worth 10 points on the next test requires them to do the derivation.
In a math methods course that I took in ed school, we had to prepare a set of lesson plans for a unit. Mine was on quadratic equations, and I mentioned my idea of allowing extra credit on a test for those who could derive the formula. The teacher felt that students would merely memorize the steps and would not understand what was really going on.
Having done this for several classes, my experience is that students who try to memorize the steps usually do not succeed. Those who understand the process of completing the square, usually fare better. Although they may miss a few steps along the way, they generally get most of it.
Today’s Warm-Ups contain some problems that front-end load some of the steps that are used in the derivation.
Warm-Ups.
Problem 1 is straightforward, though some students will still have difficulty simplifying it. Problems 2 through 4 are front-loading procedures they will use in the day’s lesson on deriving the quadratic formula. Prompt for Problem 2: “What is the coefficient of x?” “How do we find half of that?” Problem 3 is addition of fractions; since the LCD is a prompt would be: “What do you multiply c/a by to get in the denominator. Problem 4 a prompt would be break the radical into the product of two radicals as shown in the answer. They may need to be reminded that the square root of a variable to a power is the variable raised to the power divided by two. Most students will factor for Problem 5, but some will still miss that one of the solutions is x = 0.
By Way of Introduction. I go through the Warm-Ups carefully since the last three problems are procedures they will use in the derivation of the formula. I start off the lesson by having a student read aloud one of the posters I have on my wall—a quote from Rene Descarte:
“Each problem that I solved became a rule, which served afterwards to solve other problems.”
“We’ve seen this all year long. You learn a rule for solving problems and then use it in more complex problems. But this quote is most evident when it comes to deriving the quadratic formula, which is what we will do today. I mentioned that the formula that you’ve been using is the end result of solving the standard form of the quadratic equation using the method of completing the square. So with that, let us get started.”
Derivation of the Quadratic Formula. “First, as always, I want to know what the standard form is for a quadratic equation.” I call on any student. Usually all students know it by now, but if not, I’ll call on another to get the answer, reminding them that I expect them to know this just as I expect them to know the quadratic formula. I have them write this and all the steps that follow in their notebooks.
“We want to complete the square. What is usually the first thing we do?”
There are two possible answers that I will get and I’ll go with either one. One answer is to move c to the other side. The other is divide all terms by a. Assuming the first possibility is given that’s what I do; otherwise I divide all the terms by a and then move c to the other side.
Move c to the other side:
“I notice there’s a coefficient in front of the quadratic term. What do I do?”
They tell me to divide all the terms by a. “Sounds good, but since we’re dividing by a tell me what the excluded value is for a. What can it not equal?”
I’m told it’s zero and I proceed.
“What next?” There’s usually silence. “Well how have you done it in the past? What is this procedure called?” This usually jars the answer loose: Complete the square.
“Now actually you’ve already done this in your Warm-Ups. So tell me what it is I need to do.”
Someone will tell me to multiply b/a by1/2 and square it.
“Now what do I do with this?” I’m told to add it to both sides.
“I wrote the right hand side in the order I did because it will make things a bit easier in the end. So what do we do now that we have a perfect square trinomial on the left hand side?”
Someone will know that we factor it. I may need to remind them that the second term is the square root of the third term of the perfect square trinomial.
“Are things starting to look a little familiar?” Students agree. “Now let’s make the right hand side a bit more presentable. We have the sum of two fractions. We need to add them, and guess what? You already did that in your Warm-Up. Tell me what to do. In fact, come on up here and do it.” I pick a student who I know won’t mess it up—important not to break the flow here.
“Now our equation looks even more familiar.”
“What is the next step?” I do this one because I don’t want them to do two steps in one.
“This is getting exciting now. Can anyone simplify the right hand side, and yes, you did that in your Warm-Up. I think I made this too easy for you.”
Someone will tell me to make the denominator 2a since it’s the square root of 4a².
“Come on up and do it.”
“Now solve for x. Who can do it?”
“Last step: make it one fraction. Come on up!”
“And there you have it. The quadratic formula.”
Homework. I assign more quadratic equations, but this time allow them to pick the method of their choice. I also want them to work on simplifying the final answer. For example: