Lesson 3 | Quadratic Equations and Applications | 9th Grade Mathematics

Objective

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

Common Core Standards

Core Standards

The core standards covered in this lesson

A.REI.B.4.B — Solve quadratic equations by inspection (e.g., for x² = 49), 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 a ± bi for real numbers a and b.

Reasoning with Equations and Inequalities

A.REI.B.4.B — Solve quadratic equations by inspection (e.g., for x² = 49), 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 a ± bi for real numbers a and b.

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

Rewrite a quadratic equation in vertex form by completing the square in order to reveal the minimum or maximum of the function.
Solve a quadratic equation by completing the square and taking the square root.
Understand there is more than one way to algebraically determine features of quadratic functions (such as roots, vertex); some approaches are more efficient than others, depending on the structure of the equation.

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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

A quadratic function and its graph are shown below.

$${y=x^2+3x+6}$$

Estimate the vertex from the graph.
Determine the vertex using the formula $${ x=-{b\over{2a}}}$$.
Rewrite the equation in vertex form by completing the square. Identify the vertex.

Guiding Questions

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Problem 2

Rewrite the equation below in vertex form.

$${x^2+4x-3=y}$$

What is the vertex of the parabola? Is it a minimum value or a maximum value?
Use the vertex form to find the roots of the function.

Guiding Questions

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Problem 3

Solve using any approach.

a. $$ 4(x+5)(x-6)=0$$

b. $${(-2x+8)^2=36}$$

c. $${3x^2-12x-1=0}$$

Guiding Questions

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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

Write the equation below in vertex form. Then find the vertex and the roots.

$${-x^2-6x-5=y}$$

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 Free Algebra 1 Worksheets Solving Equations by Completing the Square
Mister, is this right? My Favorite: Completing the Square — See the photo of the worksheet with "Solve using any method"
Mathematics Vision Project: Secondary Mathematics Two Module 2: Structures of Expressions — Lesson 2.5 "Set"
Illustrative Mathematics Completing the Square

Lesson 2

Lesson 4

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

Reasoning with Equations and Inequalities

Seeing Structure in Expressions

Foundational Standards

Seeing Structure in Expressions

Seeing Structure in Expressions

Criteria for Success

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Problem 3

Guiding Questions

Problem Set

Target Task

Additional Practice

Lesson Map

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