QE 4 Traditional Math: Solving Quadratic Equations by Completing the Square
Eighth Grade Algebra
Having had practice with completing the square, including those expressions where the linear term is a fraction, students are now ready to put it to use. In the lesson students first learn how to solve such equations when the coefficient of the quadratic term equals 1. This is followed by equations where the quadratic term is other than 1. For both types of equations, solutions may involve fractional forms such as the following:
The above solution is not in simplified form, which I require students to be able to do. This involves writing it as one fraction, and simplifying the radical; i.e:
Some solutions may also require factoring and cancelling:
I mentioned in a previous section that although quadratic equations differ from linear equations, the foundational concepts and procedures in the earlier part of the course still play a role. Students must know the procedures for working with algebraic fractions as well as factoring. Writing algebraic expressions in the simplest way is an important skill, just as is writing sentences to be easily understood.
Nevertheless, some students find that the new information of solving quadratic equations eclipses the operations of simplification. This leaves me with some options. I can give penalizes students for not simplifying the final answer even if it is technically correct. Or partial credit for an answer that is correct but not simplified. I have chosen not to do that, but instead to indicate on the test or quiz that students can get an extra credit point for simplifying the answer. This has had the effect of motivating some students to do so.
Completing the Square to Solve Equations, when A = 1. After going through any homework problems students want to see worked, I start in: “What is the standard form for a quadratic equation” I ask and call on anyone whether I believe that person can answer or not. I expect all to answer correctly at this point given how many times I’ve asked the question. I usually get the correct answer. If not, I’ll as for a student to provide the correction.
“And as you know (I hope), the first term is called the ‘quadratic term’, the second term is the linear term, and the third term, C, is the constant. Yesterday we learned how to complete the square. We can now use this to solve quadratic equations. This allows us to solve quadratic equations for which the trinomial cannot be factored. We’ll start out with equations where the coefficient of the quadratic term is 1. Then we’ll look at equations where it might be 2, 3, or any other number but 1.” I have them copy the first worked example in their notebooks.
Examples. I start with problems that can be solved by factoring. After solving by completing the square, I ask which method they would prefer. The answer is always factoring. I do this because later they will be using the quadratic formula; I want them to be able to recognize and use the method that is most appropriate.
Solving Equations When A = Other Than 1. I remind students that for problems like the last one in the examples, they are just adding fractions at the end. Also it is important to recognize when there is a perfect square in the denominator of the radical.
“Now let’s look at equations when A is something other than 1. we divide each term by the coefficient of the quadratic term.” I write on the board:
“The first thing we do is move the -9 to the other side.”
“Now we divide every term—on both sides—by the coefficient of the quadratic term. Which is 3 in this case.
“We have a fraction coefficient for the linear term. So how do we divide by 2?”
I am not surprised if there is hesitancy—when confronted with new methods and information even the familiar seems unfamiliar. Someone will remember, however.
“I usually do this on the side of the page—maybe put it in a box to remind you of all your steps. Now add it to both sides—and keep things in fraction form. So turn 3 into a fraction.”
“Now what do we do with the left hand side?”
They tell me to factor it, and I ask someone to now write it on the board.
“What should we do with that 9 in the denominator under the radical? Finish this up in your notebooks.” I go around to see what they’ve done.
Examples. I include some from the homework assignment so we are starting in on the homework. I assure them that these seem difficult because there are a lot of steps, but just to keep careful track of what they are doing. Neatness helps!
I work through the first one with the class, labeling the steps as we go:
Homework. Problems are similar to these. They include problems where A = 1, and other than 1. I try to allow maximum time to work on these with the students during class time. I also will go through the more difficult ones the next day.