Solving Quadratic Equations with Perfect Squares

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SWBAT solve a difference of two perfect squares quadratic equation by taking square roots and factoring.

Big Idea

This lesson will serve as a precursor to the concept of completing the square.


5 minutes

Today, I will have students work with their partner on the opening question, Slide 1 of Solving_Quadratics_with_Square_Roots_Launch.  My students will quickly notice that this question is similar to one they answered in a previous lesson.  I encourage students to think back to the last time we did this question. I ask, "What's the difference today?"  If a student comes up with, "last time, the question said the word positive", we will take it from there.  If not, I will tell them that last time we were looking for a positive integer.  

Once I make sure that students are aware that they are looking for something different in their solution today, I will ask them to turn and talk to discuss how this omission changes their solution.  After a couple of minutes, I plan to bring the class back together to discuss this question as a group.  

I expect many of my students will come up with the fact that now the answer could be 3 or -3 without writing an equation. My goal for this task is to push them to explain their thinking. I will ask,"Why is this true?"  Students should explain that 3*3 is 9 and that -3*-3 is also 9.  

Teaching Note: At this point I often like to ask, "Why isn't -3^2 equal to -9?"  The evaluation of this expression is a source of a lot of mistakes by my students. I ask this question to help remind students of the different between -3 * -3 and -3^2. 

After discussing the problem as a number puzzle, in Slide 2 I ask students to demonstrate the solution using a quadratic equation. I plan to have a student write their solution on the board while other students work at their seats. I expect that many students will factor the difference of perfect squares and then ended up with two solutions by using the Zero Product Property since this has been a focus over the last several days.  

I also want students to consider a second approach: we can solve for x^2 and take the square root of each side. I want students to bear in mind the idea that the inverse operation of squaring is taking the square root.  This idea should be familiar from student's study of the Pythagorean Theorem in Grade 8.  But now, we know that when we take the square root of 9 the solution can include 3 and -3.  It is inevitable that some students will be confused as to why they have never obtained a negative solution in the past. I usually explain that taking the square root of a number to find a measurement generally requires a positive solution, sometimes called the Principal Square Root.  In this situation, however, we are working with an equation. There are two solutions because more than one numerical value makes the equation true. Since 3^2=9 and (-3)^2=9 both x = 3 and x = -3 are valid solutions.

Guided Practice

25 minutes

For today's Guided Practice, I give students Solving_Quadratics_with_Square_Roots, a worksheet with progressively sequenced practice problems. As a start, I ask students to answer the first four questions with a partner, keeping in mind the different strategies that were discussed during today's Launch. I encourage students to use more than one solution method and to verify that their solutions match for each method of solving. 

After students have had time to complete the first section, I ask all of them look at Question 5. I ask students to think of a plan, "How can you solve this equation?" Then, I'll ask students to turn-and-talk to their partners before we discuss possible strategies as a class.  If students are having difficulty answer the question, I will ask probing questions like:

  • What makes this question different from Questions #1-4?
  • Why can't I factor this into (x+2)(x-9) (or some other product of binomials)?
  • Can you estimate the answer, perhaps by thinking of other perfect squares that you do know?

I want to make sure that students are thinking critically about the equation. I want them to identify the problems that they need to solve in order to determine the solution (MP1).  If I think that it will help, I will remind my students of the work that we did in the lesson on Simplest Radical Form.  

After this discussion, the work of guiding students through a solution to Question 5 generally goes smoothly. I will solve using square roots first.  When they record their results, I ask my students to write the answer as "sqrt(18) or -sqrt(18)".  Then, I ask them to rewrite the answer in simplest radical from.  I think that this is an important step to take before showing students how to solve Question 5 by factoring.

To demonstrate a solution to Question 5 by factoring, I begin by asking my students to look for a pattern in the solution processes to Questions #1-4 (MP7).  I want my students to see that in each case, the factors contain both the positive and negative square root of the constant term (example x^2-49 = (x+7)(x=7)).  Since we have establishes that the square root of 18 can be expressed as sqrt(18), I will ask, "Can you rewrite the equation as (x-sqrt(18))(x+sqrt(18))=0?" With my students, I have found this approach to be quite successful.

After guiding the students through Question 5, I ask students solve Questions 6-8 to review and practice the technique I just showed them.  I ask for answers in simplest radical form where possible.  

Questions 9 and 10 have a similar problem structure, but they add an additional complication to consider. When my students reach these problems, I encourage them to try to use the same technique that was used for Problems 5-8. For example, in Question 9, it is possible to use the technique if (x+2)^2 is viewed as similar to (x)^2. There is a leap that students must take here, but, if they are reasoning about the structure of the problems, rather than simply applying a procedure, they should be able to cross the gap. 


10 minutes

Today's Ticket out the Door, Solving_Quadratics_with_Square_Roots_Close, will serve as a quick assessment of student progress during the lesson.  In Question #1 I ask students to solve the equation using both techniques discussed in class today.  In Question 2, students will have a choice of how to solve the equation and then write their answer in simplest radical form.