I like to introduce the quadratic formula before showing its derivation which I do the following day. I give a complete the square problem in the Warm-Ups which I then use to demonstrate that the formula gives the same answer. I then spend time showing how the quadratic formula is used.
I also go over homework problems which involved completing the square. I make a point of showing how the radical on the right hand side often has a denominator that is a perfect square, and how that serves as a shortcut. Specifically, in the day’s Warm-Up, there is a complete the square problem. At the point when they must simplify the right hand side, I point out the shortcut:
Oneof the reasons I do this is to plant that seed for the next day’s derivation of the quadratic formula.
Problem 1 will cause students some difficulty. Prompts: “Can we write a with an exponent? What is the exponent?” “When we multiply powers with the same base, what do we do with the exponents?” Problem 2 is an equation not an inequality, so students can solve as shown. Students may notice that Problem 3 has the same answer. I point out that if we square both sides of the equation in Problem 2 we get Problem 3. Problem 4 is a mixture problem which they have had before. Students may need to be reminded to use substitution for solving the two equations. Problem 5 is similar to their homework problems. I work through this one step by step.
The Quadratic Formula. I go through the answer to Problem 5 of the Warm-Ups step by step. We then go through any homework problems that cause difficulty. Generally, students have trouble working with fractions, particularly combining a whole number on the right hand side with the “c” term that is added to both sides. I advise them to write the whole number with a denominator of 1 to help them see that when determining what 2 when added to 9/16, say, they just have to multiply 2 by the desired denominator of 16 to obtain 32/2.
After finishing discussing any homework problems I announce: “Let’s look at Problem 5 of the Warm-Up again.”
“What is the standard form of the quadratic equation?” I ask this question often, so most students will know the answer and I write it under Problem 5:
“Completing the square is used for a variety of things, and one of them is that if we were to solve the standard form of the quadratic equation using it, we would obtain a formula that can be used to solve any quadratic equation. It’s called the quadratic formula. I showed it to you at the beginning of this unit. We will see tomorrow how it’s derived.”
Using the Quadratic Formula. “Let’s see how it works using Problem 5.”
We want to match up the a, b, and c values of the standard from with this equation. “The a value is the coefficient of the quadratic term. What is it in this equation?” They see that it is 2 and I write a = 2 on the board.
“How about b and c?” They identify these readily, although some will say c is 4. “The number of variable stays with the sign,” I say and I write the values on the board as well: b = 3, c = -4.
We want to match up the a, b, and c values of the standard from with this equation. “The a value is the coefficient of the quadratic term. What is it in this equation?” They see that it is 2 and I write a = 2 on the board.
“How about b and c?” They identify these readily, although some will say c is 4. “The number of variable stays with the sign,” I say and I write the values on the board as well: b = 3, c = -4.
“This is the same answer we obtained by completing the square. Which do you think is easier?”
Most will say the formula. Some will ask why we weren’t taught this in the beginning. “Good question. I want you to see how completing the square leads to this formula, which we will do tomorrow. But another reason is that you will use completing the square for other problems in algebra 2 and in pre-calculus.”
I also emphasize that when using the formula, to be careful when subtracting the 4ac term. “If c is a negative value, you will end up adding two numbers. If c is positive, you will be subtracting two numbers. That’s why I wrote it as I did with the parentheses.”
Examples. The examples are taken from the homework assignment. Some of the problems can be solved by factoring, and I tell them this. “For today’s homework you will use the quadratic formula for all problems. In the future, though, you may use whatever method you find easiest. But the formula can be used for any quadratic equation.”
Homework. The homework continues with problems of this type. In some cases, the equation will not be stated in standard form. Students must remember to put all equations in standard form in order to identify the a, b, and c values correctly. I also assign one or two word problems, one of which is below:
Two tin squares together have an area of 325 square inches. One square is 5 inches longer than the other. Find the length of the sides of each square.
I advise students that they will get two answers one of which will be negative. Since we can’t have a length that is negative, they must reject that answer.
Answer: Let x = length of side of smaller square. Then, x + 5 is length of side of the larger square. Area of a square is the length of side squared.
Students will find it easier to factor than to use the quadratic formula. If they use the formula they will need to know that the square root of 625 is 25. This is another perfect square that they should know, along with the squares from 11 through 20.