Lesson 11 | Quadratic Functions and Solutions | 9th Grade Mathematics

Objective

Solve quadratic equations by taking square roots.

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.

Foundational Standards

The foundational standards covered in this lesson

8.EE.A.2

Expressions and Equations

8.EE.A.2 — Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.

Criteria for Success

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

Solve equations by taking the square root and by factoring and showing that the solutions are the same.
Choose an efficient method to solve a quadratic equation based on the structure of the equation.
Solve and interpret a quadratic equation in context.

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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 equation is shown below.

$${x^2=(2x-9)^2}$$

a. Solve this equation by expanding and then factoring.

b. Solve this equation by taking the square root of each side.

Guiding Questions

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References

Illustrative Mathematics Two Squares are Equal

Two Squares are Equal, accessed on Sept. 25, 2017, 2:47 p.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.

Problem 2

A physics teacher put a ball at the top of a ramp and let it roll down toward the floor. The class determined that the height of the ball could be represented by the equation $${h=-16t^2+4}$$, where the height, $$h$$, is measured in feet from the ground and time, $$t$$, is measured in seconds.

What do you notice about the structure of the quadratic expression in this problem?
In the equation, explain what the $$4$$ represents.
Determine the time it takes for the ball to reach the floor. Interpret your solution(s) in context of the problem.

Guiding Questions

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References

EngageNY Mathematics Algebra I > Module 4 > Topic A > Lesson 6 — Example 1

Algebra I > Module 4 > Topic A > Lesson 6 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

Solve the equations.

a. $${4a^2=16}$$

b. $${5b^2-25=0}$$

c. $${8-c^2=5}$$

d. $${(x-2)^2=9}$$

e. $$3{(x-2)^2=9}$$

f. $${6=24(x+1)^2}$$

References

EngageNY Mathematics Algebra I > Module 4 > Topic A > Lesson 6 — Exit Ticket

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.

Include examples where the constant value is not a perfect square root and students represent the solutions as numerical expressions involving the square root symbol

Kuta Software Free Algebra 1 Worksheets Solving Quadratic Equations with Square Roots
EngageNY Mathematics Algebra I > Module 4 > Topic A > Lesson 6 — Exercises and Problem Set

Lesson 10

Lesson 12

Lesson Map

Topic A: Features of Quadratic Functions

Compare quadratic, exponential, and linear functions represented as graphs, tables, and equations.

F.IF.B.4 F.LE.A.2

Identify key features of a quadratic function represented graphically. Graph a quadratic function from a table of values.

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

Calculate and compare the average rate of change for linear, exponential, and quadratic functions.

F.IF.B.4 F.IF.B.6 F.LE.A.3

Topic B: Factoring and Solutions of Quadratic Equations

Factor quadratic expressions using the greatest common factor. Demonstrate equivalence between expressions by multiplying polynomials.

A.APR.A.1 A.SSE.A.2 A.SSE.B.3.A

Identify solutions to quadratic equations using the zero product property (equations written in intercept form).

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

Factor quadratic equations and identify solutions (when leading coefficient is equal to 1).

A.SSE.A.1.A A.SSE.B.3.A

Factor quadratic equations and identify solutions (when leading coefficient does not equal 1).

A.SSE.A.1.A A.SSE.B.3.A

Factor special cases of quadratic equations—difference of two squares.

A.SSE.A.1.A A.SSE.A.2 A.SSE.B.3.A

Factor special cases of quadratic equations—perfect square trinomials.

A.SSE.A.2 A.SSE.B.3.A

Solve quadratic equations by factoring. Compare solutions in different representations (graph, equation, and table).

A.SSE.B.3.A F.IF.C.8.A F.IF.C.9

Solve quadratic equations by taking square roots.

A.REI.B.4.B

Graph quadratic functions using $${x-}$$intercepts and vertex.

A.APR.B.3 F.IF.B.4 F.IF.C.7.A F.IF.C.8.A

Topic C: Interpreting Solutions of Quadratic Functions in Context

Interpret quadratic solutions in context.

A.CED.A.1 F.IF.B.4 F.IF.B.5 F.IF.C.8.A

Quadratic Functions and Solutions

Objective

Common Core Standards

Core Standards

Reasoning with Equations and Inequalities

Foundational Standards

Expressions and Equations

Criteria for Success

Anchor Problems

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

References

Problem Set

Target Task

References

Additional Practice

Lesson Map

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