Quadratic Equations and Applications

Lesson 15

Math

Unit 8

9th Grade

Lesson 15 of 15

Objective


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

Common Core Standards


Core Standards

  • A.REI.C.7 — Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x² + y² = 3.
  • A.REI.D.11 — Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Foundational Standards

  • 8.EE.C.8
  • A.REI.C.6

Criteria for Success


  1. Understand that a system of linear and quadratic equations can have two, one, or no solutions; identify graphically examples of each case. 
  2. Interpret the algebraic solution of a system of linear and quadratic equations as representing two, one, or no solutions. 
  3. Solve real-world applications of systems of linear and quadratic equations.
Fishtank Plus

Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.

Anchor Problems


Problem 1

Three systems of linear and quadratic equations are shown below. 

a. How many solutions are there to each system?

b. Verify algebraically the number of solutions. Give the coordinate points of the solutions. 

Guiding Questions

Create a free account or sign in to access the Guiding Questions for this Anchor Problem.

Problem 2

Use the system of equations below to answer the questions that follow. 

$${{g(x)}=x^2-8x+12}$$

$${h(x)={1\over2} x-7}$$

  1. Does the system have any solutions? Justify your answer. If yes, name the coordinate point(s). 
  2. Does the quadratic function, $${g(x)}$$, have any solutions? Justify your answer. If yes, name the coordinate point(s).

Guiding Questions

Create a free account or sign in to access the Guiding Questions for this Anchor Problem.

Problem 3

A ball is thrown vertically upward from a second-floor balcony, 15 feet above the ground, at an initial velocity of 50 feet per second. 

At the same time, a remote-controlled toy plane takes off from the third-floor balcony, 30 feet above the ground, and rises 10 feet every second. 

  1. Write a function, $${b(x)}$$, to represent the height of the ball after $$x$$ seconds.
  2. Write a function, $$p(x)$$, to represent the height of the toy plane after $$x$$ seconds. 
  3. Will the ball and the plane ever be at the same height at the same time? If so, how many times will this happen and at what height(s)? 

Guiding Questions

Create a free account or sign in to access the Guiding Questions for this Anchor Problem.

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


Problem 1

How many solutions are there to the system below? Justify your answer.

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

$${y=x-8}$$

Problem 2

What is the solution to the system below?

$${y=(x+1)^2-2}$$

$${y=-2}$$

icon/arrow/right/large copy

Lesson 14

Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Deriving the Quadratic Formula

Topic B: Transformations and Applications

Request a Demo

See all of the features of Fishtank in action and begin the conversation about adoption.

Learn more about Fishtank Learning School Adoption.

Contact Information

School Information

What courses are you interested in?

ELA

Math

Are you interested in onboarding professional learning for your teachers and instructional leaders?

Yes

No

Any other information you would like to provide about your school?

Effective Instruction Made Easy

Effective Instruction Made Easy

Access rigorous, relevant, and adaptable math lesson plans for free