Lesson 2 | Quadratic Equations and Applications | 9th Grade Mathematics

Objective

Complete the square.

Common Core Standards

Core Standards

The core standards covered in this lesson

A.SSE.B.3.B — Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

Seeing Structure in Expressions

A.SSE.B.3.B — Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

Foundational Standards

The foundational standards covered in this lesson

A.SSE.A.2

Seeing Structure in Expressions

A.SSE.A.2 — Use the structure of an expression to identify ways to rewrite it. For example, see x<sup>4</sup> — y<sup>4</sup> as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).
A.SSE.B.3.A

Seeing Structure in Expressions

A.SSE.B.3.A — Factor a quadratic expression to reveal the zeros of the function it defines.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Create a perfect square trinomial within a quadratic expression.
Understand the process for completing the square when the leading coefficient is equal to $$1$$, when the leading coefficient is not equal to $$1$$, when the value of $$b$$ is even, and when the value of $$b$$ is odd.
Understand how to maintain the equivalence between expressions or the balance within an equation.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

In terms of pacing, this lesson may be extended over 2 days to adequately cover all of the different variations of expressions and equations. Students will have time to practice and apply this skill of completing the square in Lessons 3–5.
The approach taken in this lesson focuses on an algebraic method versus a geometric method. If you would like to take a geometric approach, the Desmos activity Introducton to Completing the Square is a great resource.
This lesson focuses on the process of completing the square. In Lesson 3, students will apply this process to find the vertex and to solve for the roots.

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

Complete the table with the missing values in order to "complete the square."

$${x^2+6x+9}$$	$${{{(x+}}3{{{)^2}}}}$$
$${x^2+8x+16}$$
$${x^2-10x+25}$$
	$${{(x-}7{{{)^2}}}}$$
$${x^2-20x+}$$_______	$${(x-}$$_______$${{{)^2}}}$$
$${x^2+16x+}$$______	$${{(x+}}$$_______$${{{)^2}}}$$
$${x^2+bx+}$$______	$${{(x+}}$$_______$${{{)^2}}}$$

Guiding Questions

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References

f(t) Completing the Square

Completing the Square by Kate Nowak is made available on Function of Time under the CC BY-NC-SA 3.0 license. Accessed July 12, 2018, 4:32 p.m..

Problem 2

Complete the square in each expression below to include a perfect square binomial.

a. $${x^2+6x+11}$$

b. $${x^2+5x-10}$$

Guiding Questions

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Problem 3

Rewrite each equation into vertex form by completing the square.

a. $${x^2-12x+5=y}$$

b. $${2x^2+16x+3=y}$$

Guiding Questions

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References

EngageNY Mathematics Algebra I > Module 4 > Topic B > Lesson 12 — Example 1

Algebra I > Module 4 > Topic B > Lesson 12 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.

Problem Set

A set of suggested resources or problem types that teachers can turn into a problem set

Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

What is the minimum value taken by the expression $${{{(x-}}4)^2+6}$$? How does the structure of the expression help to see why?
Rewrite the quadratic expression $${ x^2-6x-3}$$ in the form $${{(x-}}$$ _____$${{ )^2+}}$$ ______ .
Rewrite the quadratic expression $${-2x^2+4x+3}$$ in the form ____ $${{(x-}}$$ _____$${{ )^2+}}$$ ______ .

References

Illustrative Mathematics Rewriting a Quadratic Expression

Rewriting a Quadratic Expression, accessed on July 12, 2018, 11:55 a.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Modified by Fishtank Learning, Inc.

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

Kuta Software Completing the Square
Mathematics Vision Project: Secondary Mathematics Two Module 2: Structures of Expressions — Lesson 2.3 "Building the Perfect Square" (Note, this is a geometric approach) and "Set"
Mathematics Vision Project: Secondary Mathematics Two Module 2: Structures of Expressions — Lesson 2.4 "Set"
EngageNY Mathematics Algebra I > Module 4 > Topic B > Lesson 11 — Problem Set (Note: these problems all include a leading coefficient of 1)
EngageNY Mathematics Algebra I > Module 4 > Topic B > Lesson 12 — Exercises and Problem Set (Do not include the application problems. These include problems with a leading coefficient not equal to 1)

Lesson 1

Lesson 3

Lesson Map

Topic A: Deriving the Quadratic Formula

Describe features of the vertex form of a quadratic function and write quadratic equations in vertex form from graphs.

A.SSE.B.3 F.IF.B.4 F.IF.C.8

Complete the square.

A.SSE.B.3.B

Complete the square to identify the vertex and solve for the roots of a quadratic function.

A.REI.B.4.B A.SSE.B.3.B

Solve and interpret quadratic applications using the vertex form of the equation.

A.SSE.B.3.B F.IF.C.8.A

Convert and compare quadratic functions in standard form, vertex form, and intercept form.

F.IF.B.4 F.IF.C.9

Derive the quadratic formula. Use the quadratic formula to find the roots of a quadratic function.

A.REI.B.4.A

Determine the number of real roots of a quadratic function using the discriminant of the quadratic formula.

A.REI.B.4.B F.IF.C.7.A

Graph quadratic functions from all three forms of a quadratic equation.

F.IF.C.7.A

Topic B: Transformations and Applications

Describe transformations to quadratic functions. Write equations for transformed quadratic functions.

F.BF.B.3

Graph and describe transformations to quadratic functions in mathematical and real-world situations.

F.BF.B.3

Write and analyze quadratic functions for projectile motion and falling bodies applications.

A.CED.A.2 F.IF.C.8.A F.IF.C.9

Write and analyze quadratic functions for geometric area applications.

A.CED.A.2 F.IF.C.8.A

Write and analyze quadratic functions for revenue applications.

A.CED.A.2 F.BF.A.1.B F.IF.C.8.A

Solve and identify solutions to systems of quadratic and linear equations when two solutions are present.

A.REI.C.7 A.REI.D.11

Solve and identify solutions to systems of quadratic and linear equations when two, one, or no solutions are present.

A.REI.C.7 A.REI.D.11

Quadratic Equations and Applications

Objective

Common Core Standards

Core Standards

Seeing Structure in Expressions

Foundational Standards

Seeing Structure in Expressions

Seeing Structure in Expressions

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

Problem 3

Guiding Questions

References

Problem Set

Target Task

References

Additional Practice

Lesson Map

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