Unit 6, Lesson 13
Completing the Square and Complex Solutions
Integrated Math 2
13.1
Warm-up
Standards Alignment
Building On
Addressing
Building Toward
HSA-REI.B.4.b
Solve quadratic equations by inspection (e.g., for ), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as for real numbers and .
Instructional Routines
NoneMaterials
NoneActivity Narrative
This Warm-up prompts students to connect quadratic equations in standard form, factored form, and their solutions. Up until this point, students have solved quadratic equations with only real solutions. This idea is key for upcoming activities where students solve quadratic equations that have no real solutions but do have non-real solutions.
Launch
NoneActivity
NoneMatch each equation in standard form to its factored form and its complex solutions.
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5, -5
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Student Response
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Building on Student Thinking
Activity Synthesis
Select students to share how they matched equations and solutions. The important idea to discuss is that the kind of numbers that can be solutions to these equations has expanded to include non-real complex numbers.
13.2
Activity
Instructional Routines
NoneMaterials
NoneActivity Narrative
The purpose of this activity is for students to solve quadratic equations of the sort that result from completing the square, including equations that have solutions that involve imaginary numbers. The left-hand side of each equation uses the same expression so that students can focus on what taking the square root does in each case. In this way, students are primed to make use of structure to see how imaginary numbers arise when solving quadratic equations (MP7).
13.3
Activity
Instructional Routines
MLR2: Collect and DisplayMaterials
NoneActivity Narrative
The purpose of this activity is for students to solve two related quadratic equations by completing the square. The process of solving will be nearly identical, with the difference being that one will involve square roots of a negative number and the other will not. It is not necessary for students to prove a general statement about all monic quadratic equations, but it is important for students to compare and understand why some have real solutions and others have non-real solutions.
Launch
Engagement: Develop Effort and Persistence. Provide tools to facilitate information processing or computation, enabling students to focus on key mathematical ideas. For example, allow students to use calculators to support their reasoning.
Supports accessibility for: Memory, Conceptual Processing
Supports accessibility for: Memory, Conceptual Processing
Activity
None13.4
Activity
Optional
Instructional Routines
MLR8: Discussion SupportsMaterials
To Gather
Graphing technologyActivity Narrative
The purpose of this activity is to connect graphs of quadratic functions with solutions to related quadratic equations to describe how many solutions an equation has, and whether those solutions are real or non-real. Students have already solved two of the equations in the previous activity, and the equation they haven’t solved involves the perfect square that connects all three of the equations. Graphically, intersects the -axis twice, which corresponds to the two real solutions to . The graph intersects the -axis once, which corresponds to the single real solution to . However, the graph doesn’t intersect the -axis, which reflects the fact that the complex solutions to can’t be described with a real number line.
Lesson Synthesis
The purpose of this discussion is for students to describe some conditions that show the number and type of solutions for quadratic equations being solved with completing the square. Here are some questions to consider:
- “How do you know when imaginary numbers will show up when you are solving a quadratic equation?” (Imaginary numbers show up when the process of solving requires finding a number that squares to make a negative number. For example, with , the quantity is a number that squares to make -3, so .)
- “Without completely solving, how many solutions does have? Are they real solutions or do they involve imaginary numbers?” (It has two real solutions. One way to see this is to recognize that it is in vertex form, and the graph of the function is upward facing with a vertex below the -axis. That means the graph will intersect the -axis in two places, which corresponds to two real solutions.)
- “What about ?” (It has two non-real solutions. One way to see this is to subtract 5 from each side and notice that a quantity squares to make a negative number. Every negative number has two imaginary square roots, so there will be two non-real solutions.)
- “What about ?” (There will be one real solution, which is 3. The structure of the expression reveals that substituting 3 for will produce a value of 0, and squaring any value other than 0 doesn’t produce 0.)
- “Is it possible for a quadratic equation to have only 1 non-real solution?” (Yes. For example, the equation has exactly one solution, which is .)
Student Lesson Summary
Sometimes quadratic equations have real solutions, and sometimes they do not. Here is a quadratic equation with equal to a negative number (assume is positive):
This equation has imaginary solutions and . By similar reasoning, an equation of the form:
has non-real solutions if is positive. In this case, the solutions are and .
It isn’t always clear just by looking at a quadratic equation whether the solutions will be real or not. For example, look at this quadratic equation:
We can always complete the square to find out what the solutions will be:
This equation has non-real, complex solutions and .