Unit 4, Lesson 16
Solving Quadratics (Optional)
Algebra 2
16.1
Warm-up
Standards Alignment
Building On
Addressing
HSA-REI.B.4.a
Use the method of completing the square to transform any quadratic equation in into an equation of the form that has the same solutions. Derive the quadratic formula from this form.
Instructional Routines
NoneMaterials
NoneActivity Narrative
In this Warm-up, students must decide if each quadratic expression is a perfect square. This review of perfect squares will be helpful later in the lesson when students complete the square to solve various quadratic equations.
Launch
NoneActivity
NoneThe expression is equivalent to . Which expressions are equivalent to for some number ?
Student Response
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Building on Student Thinking
Activity Synthesis
Select a few students to explain which expressions are perfect squares and how they know. The important idea to review is that the binomial squares to become , so in order for such a quadratic expression to be a perfect square of a binomial, its constant must be the square of half the coefficient of .
16.2
Activity
Instructional Routines
MLR8: Discussion SupportsMaterials
NoneActivity Narrative
The purpose of this activity is for students to analyze two valid methods of solving a quadratic equation (completing the square and the quadratic formula) that result in equivalent answers that look different. This gives students the opportunity to review the two methods. Students make viable arguments and critique the reasoning of others as they inspect the logic of each method and decide whether the different-looking answers are equivalent (MP3).
Launch
Arrange students in groups of 2. Encourage students to explain their reasoning to their partner as they work, and if there is disagreement, they should work to reach agreement. Remind students of the meaning of notation. For example, a result of after using the quadratic formula means the original equation has two solutions: and .
16.3
Activity
Instructional Routines
NoneMaterials
NoneActivity Narrative
This activity gives students the opportunity to review solving quadratic equations. Look for students who use different strategies to compare during discussion.
Launch
Arrange students in groups of 2. Encourage students to share their responses with their partner, and if there is disagreement, to work to reach agreement. Encourage students to use a different method than their partner uses on each question and to verify that the solutions are equivalent.
Action and Expression: Internalize Executive Functions. To support development of organizational skills in problem-solving, chunk this task into more manageable parts. For example, invite groups to agree on 3 of the 6 equations to solve.
Supports accessibility for: Organization, Attention
Supports accessibility for: Organization, Attention
Activity
NoneLesson Synthesis
In this lesson, students reviewed how to complete the square and use the quadratic formula to solve quadratic equations. Here are some questions for discussion:
- “What are some strategies for solving quadratic equations?” (Reasoning about values, completing the square, and the quadratic formula.)
- “What conditions about a quadratic equation would be best for each solution strategy?” (If there is no linear term, like , reasoning about the values is very efficient. If the coefficient of the quadratic term is 1 and the coefficient of the linear term is even, then completing the square is efficient. In other situations, the quadratic formula is useful.)
- “How does a quadratic equation need to be set up to use each of the strategies?” (To reason about the values, it is helpful to get the equation into the form for some number . To complete the square, it is helpful to have the equation in the form for some numbers and . To use the quadratic formula, the equation must be in the form .)
- “If one person used one method to get solutions and another person used a different method to get , how could you determine whether their solutions are actually the same?” (For the fraction, dividing gives . The 5 is the same as the first set of solutions, so they have exactly the same solutions as long as is equal to . In fact, they are equal because they are both positive numbers that square to make 20. I can tell by squaring them: and .)
Student Lesson Summary
Consider the quadratic equation . One way to solve equations like this is by completing the square.
To complete the square, note that the perfect square is equal to . Compare the coefficients of in to our expression to see that we want , or just .
This means the perfect square is equal to , so adding to each side of our equation will give us a perfect square.
The two numbers that square to make are and , so:
which means the two solutions are:
Now let’s look at another quadratic equation .
We could divide each side by 3 and then complete the square like before, but let’s use the quadratic formula:
To use this formula, we first need to put the equation in standard form and identify , , and . Rearranging, we get:
so , , and . We have to be careful to pay attention to the negative signs. Using the quadratic formula, we get:
Evaluating these solutions with a calculator gives decimal approximations -0.281 and 0.948.