Curriculum  /  Math  /  9th Grade  /  Unit 8: Quadratic Equations and Applications  /  Lesson 15

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.
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Anchor Problems

25-30 minutes

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

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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

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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

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Problem Set

15-20 minutes

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Target Task

5-10 minutes

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?

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

$${y=-2}$$

Lesson Map

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

Topic A: Deriving the Quadratic Formula

Topic B: Transformations and Applications

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