QE 1 and 2 Traditional Math: Quadratic Equations and Square Root Method

Eighth Grade Algebra

Quadratic equations for me are the pinnacle of the algebra 1 course. Typically it is taught last at a time when the end of school is near and students view every new topic like so much extra paraphernalia to cram into an overstuffed suitcase. That is why I make every effort to teach the unit on quadratic equations no later than April of the school year when learning new things still matters.

Prior to teaching this unit, and after the unit on systems of equations, students learn about radicals—an essential topic to know before venturing into the world of quadratic equations. Students learn the basic properties of square roots:

I have heard it said that algebra courses build on topics previously learned, but when it comes to quadratic equations, this is no longer true. This idea arises out of the misconception that prior to quadratics, students have been learning about linear equations, inequalities, and systems of equations and that quadratic equations don’t link up to students’ experience with linearities. While quadratic equations are solved differently than linear, that may be true, they have also learned about manipulation of algebraic expressions, factoring and squaring binomials, and algebraic fractions—all of which are necessary to work with quadratic equations.

In this unit, students may become confused over the various methods shown to solve quadratic equations. I like to emphasize that if an equation can be solved by factoring, then they should do so. Ultimately they will learn that all equations can be solved by using the quadratic formula. For some equations, however, it is easier to simply rely on factoring. For example, the following equations are easier solved using factoring, than using the quadratic formula:

Similarly, the square root method is the easiest method for other equations such as these:

Problem 1 requires students to understand what a fractional exponent represents, and how to evaluate numbers raised to a fractional exponent.  Problem 2 requires students to be able to simplify radicals by determining factors of the number under the radical that are perfect squares. In Problem 4, some students need to be reminded to put the equation in standard form; i.e., right side is zero. Problems 4 and 5 lead in to this lesson; students must be familiar with factoring as a means to solve quadratic equations.

Q2: Square Root Method of Solving Quadratic Equations.  Having gone over the Warm-Ups, I’ve reminded students that Problems 4 and 5 are quadratic equations that are solved by factoring.

“But sometimes you will have equations that cannot be factored. In this unit you will learn how to solve any and all quadratic equations, whether they are factorable or not. And in the end, you will be using a formula that looks like this,” I say and write the quadratic formula on the board:

I have to say that I take no small pleasure in the reactions that students evince, including “No way!”, “What is that?”, “Have mercy!” and “What are you trying to do to us?” to name a few.

This is the first time they have seen anything like this, and even though they have seen the plus or minus sign, the whole formula is formidable to them. I find this preview of things to come at the end of the unit far more effective than a daily objective written on the board which goes largely ignored.

“Believe it or not, when we get to this lesson, the formula will not seem as bad as it does to you now, and you will be using it to solve quadratic equations. So let’s take this one step at a time.”

“Just to see if you remember, who can tell me what the square root of x² is?”

Most students will recall this from the previous unit: x.

Moving on, I write on the board the standard form for a quadratic equation.

“In Problem 4 of today’s Warm-Up, you put the equation in standard from. When you did so, you had an equation where B equals zero. You solved it by factoring the difference of two squares. But now that you’ve had experience with square roots, we can look at this problem another way. Rather than putting it in standard form, we can take the square root of both sides.”

“We know what the square root of 4x² is; I hope. Someone tell me.”

I hear “2x”

“Good. Now what’s the square root of 81? And you’ll notice I put the plus or minus sign in front of the radical. This is because we want to know both roots of 81.”

I want to hear 9 and -9 which I generally hear. “We want to take both positive and negative roots because both will satisfy the equation and we want all solutions for the equation. So now our solution is plus or minus 9/2. That is, 9/2 and -9/2. This is the same answer you obtained by factoring is it not?”

General agreement follows.

“I hear you wondering why you can’t just factor it like you did for the Warm-Up. Yes, you can. But if I asked you solve this equation you’d have a harder time with factoring.”

“We take the square root of each side and we get this.”

“Who can simplify this radical?”  I wait for hands to go up—typically some students still find simplification of radicals difficult, but 12 is relatively easy to find a perfect square factor.

Also typically, most students forget the plus or minus sign. One way around this is to write two answers, one with a negative sign in front, and the other not. But I want them to get used to the plus or minus sign.

“In solving this problem you used the ‘Property of square roots of equal numbers’.” 

I emphasize the term “real numbers” in the above.  “If we have a negative number under the radical, it is not a real number as we discussed previously.  The square root of -12 has no real number solution.” I tell them that in algebra 2 they will learn about “imaginary numbers” as a solution. “But they are not part of the set of real numbers. You don’t need to worry about that now. In this course we will only work with real numbers.”

There is a somewhat collective sigh of relief at this, though there generally are some students who are genuinely curious—and even some who know about imaginary numbers usually because of an older sibling.

“Good; now we want to get rid of the -2 on the left hand side, so do it the way you always do it.”  I will see various forms for the answer.

“The second way is the best way to write it; the plus or minus comes afterward. The plus or minus symbol is just a short way of writing the following.”

“So what are our two answers?”  I hear 4, and 0.

We continue on, via worked examples and independent examples.

I explain that the answer can be left this way, which is called “radical form”. If I wanted a numerical answer, I would allow them to use the calculator to find the square root of 7. I want them to get used to seeing answers in this form, however, since they will be seeing it more and more. Also, when they work with the quadratic formula, it will not seem so foreign.

Homework. The homework is a continuation of these types of problems. Some of them are like Problem 5 above, so students need to be able to recognize perfect trinomial squares. Answers may be left in radical form.