This particular lesson builds on the previous one on the square root method. Since equations in the form (ax + b)² = c, can be solved by taking the square root of each side. The process of completing the square allows for an equation to be transformed into this form.
This lesson focuses only on how to complete the square. It provides a preview of how the process can be used to solve equations. The full treatment is presented in the following lesson. Although some textbooks (including Dolciani’s which I used in teaching Algebra 1) incorporate both in one lesson, I feel it is best to split up the lessons.
The method of completing the square also includes writing the perfect square trinomial in factored form—e.g., x² - 6x + 9 can be written as (x – 3)². These processes represent much new information to be absorbed and mastered. Requiring students to then solve equations by completing the square is an additional source of new information and potential confusion. Also, students are learning to work with solutions in new formats that are largely foreign. The equation 3x² -2x -9 = 0, for example, has the following solution:
Getting used to this form makes working with the quadratic formula easier to work with. The quadratic formula is what this is all leading up to.
Some students will omit the plus or minus sign, so I make a point of working through this one to make sure they know to do so. Problem 2 requires students to identify a perfect trinomial square. Also, I make a point of showing that the factored form can be a binomial squared which is part of the day’s lesson. Problem 3 requires the square root method as well as simplifying the radical. Students still struggle with simplifying radicals, so I try to keep such problems in front of them. Problems 4 and 5 review the procedure for squaring a binomial which hopefully they know how to do without resorting to actually multiplying the two binomials. The problems serve to lead in to the lesson on completing the square.
Unfactorable Trinomials. Before my student’s recall of yesterday’s lesson quickly fades, I refresh my students’ memories of what we did.
“You may recall that yesterday we had some problems that had a trinomial square as one member and a nonnegative number on the other side: For example.
“You then took the square root of each side. You also had some problems where the left hand side was a perfect square trinomial.”
“How did you solve it?”
Someone will tell me the process.
“Now if you didn’t happen to recognize it was a perfect square trinomial, you might have put it in standard form, by subtracting 9 from each side, like this.”
“Can this be solved by factoring?” I call on someone to do the problem at the board.
“This problem can be solved either way. By the square root method or by factoring. Some of you may be wondering if you need to use the square root method for some problems, and factoring for others. No, you can use whatever method works. The square root method is good method for problems where you can’t factor as we saw for Problem 3 of the Warm-Ups.
“But sometimes we have a problem that is not in the form where we can use the square root method. For example, this problem.”
“Can this be factored?” There is a general consensus that it cannot.
“No, it can’t. And it isn’t a perfect square trinomial. If only it were,” I say with a wistful sigh. “But wait! There is a way to make this into a perfect square trinomial through a process you will learn today.”
“First, let’s add 7 to both sides.” I have them write this in their notebooks.
“You’ll notice I left some space between the 2x and the equals sign. There’s a reason for that. Let’s add 1 to each side.”
“Anyone recognize the left hand side?”
I’m counting on someone to recognize that it’s a perfect trinomial square. Usually one or two students does, but if not, I’m willing to draw the next card as it were.
“So that means we can write this equation like this:”
“And now we can use the square root method which I’d like all of you to do.” I walk around checking notebooks, and pick a student to write the solution on the board.
Completing the Square. “You may be wondering how I knew to add 1 to each side. I did it through a procedure called “completing the square” which you will now learn.”
“I’m going to write the squares of some binomials. The squares of binomials are perfect square binomials, and there will be a pattern. I’m going to use the rule for squaring binomials, which is square of the first term, plus two times the product of the two terms, plus the square of the last.” After the first one, I have students tell me the steps to follow.
“You may have noticed the pattern here. In each case, the third term is the square of half the coefficient of the middle term—which is called the ‘linear term’. For example in the first one, the middle term is 2 × 5, commonly known as 10. Half of ten is…?”
I hear “five”.
“And five squared is…”
I hear 25.
“Let’s write out the steps in your notebook. We complete the square for x² + 6.”
Because the middle term of the trinomial is negative, the second term in the factored binomial will be negative.
Homework. Problems require completing the square as shown in the examples. Problems include linear terms for which the coefficients are fractions.