Lesson 7 | Quadratic Equations and Applications | 9th Grade Mathematics

Objective

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

Common Core Standards

Core Standards

The core standards covered in this lesson

F.IF.C.7.A — Graph linear and quadratic functions and show intercepts, maxima, and minima.

Interpreting Functions

F.IF.C.7.A — Graph linear and quadratic functions and show intercepts, maxima, and minima.

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

Understand why the discriminant determines the number of real roots of a quadratic function.
Use the discriminant to determine if a quadratic function has 0, 1, or 2 real roots.
Identify whether a quadratic function shown in a graph has 0, 1, or 2 real roots and describe what this means about the discriminant of the function.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

Students are not expected to define imaginary roots or to represent $${\sqrt{-1}}$$ as $$i$$. This will come in Algebra 2.

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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

To the right is a graph of a quadratic function.

How many roots does this quadratic function have? Explain your reasoning.
The equation for the function is $${ f(x)=x^2+4x+5}$$. Find the roots using the quadratic formula. How does this support what you observed in the graph?

Guiding Questions

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Problem 2

Use the quadratic formula to find the roots for each equation below.

a. $${-2x^2+x+15=y}$$

b. $${8x^2+6x+5=y}$$

c. $${3x^2-6x+3=y}$$

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

Problem 1

How many zeros does the function $${f(x)=3x^2+6x+2}$$ have? Explain your resoning.

References

EngageNY Mathematics Algebra I > Module 4 > Topic B > Lesson 15 — Exit Ticket, Question #3

Algebra I > Module 4 > Topic B > Lesson 15 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 2

Write the equation of a quadratic function with no real roots.

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.

EngageNY Mathematics Algebra I > Module 4 > Topic B > Lesson 15 — Exercises and Problem Set
Kuta Software Free Algebra 2 Worksheets Understanding the Discriminant

Lesson 6

Lesson 8

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

Interpreting Functions

Reasoning with Equations and Inequalities

Foundational Standards

Expressions and Equations

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Problem Set

Target Task

Problem 1

References

Problem 2

Additional Practice

Lesson Map

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