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# Solving Quadratic Equations

Lesson 9 of 16

## Objective: SWBAT Assess the structure of a quadratic equation and then select and carry out a solution strategy (factoring, completing the square, or using the Quadratic Formula).

## Big Idea: Its important to be flexible in solving quadratic equations. Some equations lend themselves to factoring or completing the square while others are best tackled with the Quadratic Formula.

*90 minutes*

#### Warm-Up

*15 min*

As a warm-up, students complete a set of 4 problems Warm-up: Solving Quadratic Equations. These problems are different from the previous day's work in that we are no longer only interested in what makes the function's output zero, but similar in that we are setting a quadratic expression equal to a constant.

Students work in their table groups while I check homework with the homework rubric. If students have questions about the homework or about the previous day's stations, I leave some time to answer these.

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#### Direct Instruction

*20 min*

In the previous day's lesson, we solved quadratic equations in order to find the x-intercepts of a quadratic function. In this lesson, we generalize the process of solving a quadratic equation and work on developing fluency with this process. In the ** Aligning to Assessments: Do your Homework! **reflection I discuss my planning for this lesson while showing some 2014 sample items released by the PARCC consortium that assess students' understanding of quadratic functions and equations.

I begin this part of the lesson by referring to the final exercise in the warm-up (see f(x)=g(x)), determining the value(s) of x that make the output of the quadratic function f(x)=(x-2)^{2}-6 equal to the output of the linear function g(x)=-x+2. In the warm-up activity they solved this system graphically and I ask if anyone is able to come up with a way to solve it algebraically [MP2]. Students may recognize this as a system of equations, but if not I ask more questions to lead them into this line of thinking. We will eventually end up setting f(x)=g(x) and therefore (x-2)^2-6=-x+2. I explain that now we have a quadratic equation in one variable that we have tools to solve (factoring, completing the square, Quadratic Formula).

We discuss the form this equation must be in so that factoring or the Quadratic Formula can be used and how to get it into that form. We then solve the equation using one of these methods and discuss how to check the answer. The final step in this initial exploration is to reconcile the answer we obtained algebraically with the answer we obtained graphically during the warm-up.

After answering questions, I will provide students with another pair of functions that we can use to complete the process described above. Although I am still anchoring the discussion of solving quadratic equations to quadratic functions, we are now working with non-linear systems as a bridge between quadratic functions and quadratic equations [MP7].

Finally, I present students with an equation like (x-3)^{2 }- 6 = 2x + 5. I explain solving this equation is equivalent to asking what values of x would make the output of g(x) = (x-3)^{2 }- 6 equal to the output of f(x) = 2x + 5. I introduce equations this way to help students see the connection between functions and equations.

We do several examples together of solving quadratic equations. In each case I ask students whether factoring and using the **Zero-Product Property**, completing the square, or using the **Quadratic Formula** seems most efficient. Students will not agree on the best method (because there is no one best way!) so I ask for a volunteer to solve the equation one way while I solve it another way. If we don't get the same answer we work together to figure out where the error is [MP3].

#### Resources

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#### Practice and Reinforcement

*40 min*

To practice solving quadratic equations, I use Solving Quadratics 3 Ways so that students are expected to use each method. Many students see the efficiency of using the Quadratic Formula and then begin using it for every quadratic equation but I stress the importance of staying flexible.

Students work with their table partners to complete this task, asking their peers for help before turning to the answer key or to me [MP1]. I circulate during this time and use the 3-Cup System to let me know when students are getting frustrated.

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#### Exit Ticket and Assignment

*15 min*

To determine if students are able to solve using all three methods, I send Quick Polls- Solving Quadratic Equations as an exit ticket through the Navigator system [MP5]. I ask students to solve the first equation by factoring, the second by completing the square and the third by using the Quadratic Formula. I make note of which method needs the most reinforcement (likely completing the square) to that I can provide more practice when we get to imaginary numbers, later in the unit.

For homework, I assign a puzzle. I like to use puzzles when a specific skill (like solving quadratic equations) requires fluency [MP6]. One good source is a website assembled by a retired math Colorado math teacher, Mr. Plecher.

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- LESSON 1: Introduction to Polynomials
- LESSON 2: Seeing Structure in Expressions - Factoring GCF's and Quadratics
- LESSON 3: Connecting Polynomials to Sequences
- LESSON 4: Connecting Polynomials to Geometric Series
- LESSON 5: Quadratic Functions: Standard and Intercept Forms
- LESSON 6: Quadratic Functions: Vertex Form
- LESSON 7: Flexibility with Quadratic Functions
- LESSON 8: Connecting Quadratic Functions and Quadratic Equations
- LESSON 9: Solving Quadratic Equations
- LESSON 10: Quadratic Performance Task
- LESSON 11: Quadratic Modeling (DAY 1)
- LESSON 12: Quadratic Modeling (DAY 2)
- LESSON 13: Quadratic Modeling (DAY 3)
- LESSON 14: Quadratic Modeling (DAY 4)
- LESSON 15: Review Workshop: Polynomial Functions and Expressions
- LESSON 16: Unit Assessment: Polynomial Functions and Expressions