Lesson Plan: Solving Quadratic Equations Using the Quadratic Formula

Lesson Summary

In this 50-minute lesson, students will learn to solve quadratic equations using the quadratic formula and analyze the discriminant to determine the type of solutions (real or complex). The lesson includes multimedia resources from Media4Math.com, an interactive warm-up, hands-on practice solving equations, and a quiz with an answer key. Students will use engaging drag-and-drop activities to reinforce their understanding of the quadratic formula and the discriminant.

Lesson Objectives

• Solve quadratic equations using the quadratic formula.
• Analyze the discriminant to determine the type of solutions (real or complex).
• Apply the quadratic formula to real-world scenarios.
• Develop numerical reasoning through multimedia and interactive activities.

Common Core Standards

• HSA.REI.4b: Solve quadratic equations in one variable.

Prerequisite Skills

• Substitution
• Simplifying radicals

Key Vocabulary

• Quadratic Equation: A quadratic function set equal to zero to find the roots of the equation.
  • Multimedia Resource: https://www.media4math.com/library/22143/asset-preview
• Quadratic Formula: A formula that provides the solutions to a quadratic equation, given by √x = (-b ± √(b² - 4ac))/2a. 
  • Multimedia Resource: https://www.media4math.com/library/74569/asset-preview
  • Multimedia Resource: https://www.media4math.com/library/22256/asset-preview
• Discriminant: The part of the quadratic formula under the square root (b² - 4ac) that determines the nature of the solutions. 
  • Multimedia Resource: https://www.media4math.com/library/74577/asset-preview
  • Multimeida Resosurce: https://www.media4math.com/library/74581/asset-preview
• Solutions: The values of the variable that satisfy the quadratic equation. 
  • Multimedia Resource: https://www.media4math.com/library/74592/asset-preview

Additional Multimedia Resources

• This is a collection of video definitions related to Quadratics: https://www.media4math.com/MathVideoCollection--QuadraticsDefinitions 
• This is a collection of definitions related to Quadratics: https://www.media4math.com/Definitions--Quadratics 

Warm Up Activities

Choose from one or more of these activities.

Activity 1: Slide Show

Students will use the provided slideshow to review the basics of quadratic equations and familiarize themselves with visual representations of parabolas and key terms.

• Access the slideshow: https://www.media4math.com/library/slideshow/tutorial-quadratic-functions-and-equations
• Teacher displays the slideshow on a projector. Students take notes on key concepts like the vertex, axis of symmetry, and solutions.
• Discuss: What do you notice about the shape and symmetry of the parabolas? How might this relate to solving equations?

Activity 2: Math Game

In this activity play this Interactive Math Game: Understanding the Discriminant.

URL: https://www.media4math.com/library/4833/asset-preview

Before playing the game review the definition of the discriminant.

• Discriminant: The part of the quadratic formula under the square root (b² - 4ac) that determines the nature of the solutions. 
  • Multimedia Resource: https://www.media4math.com/library/74577/asset-preview
  • Multimeida Resosurce: https://www.media4math.com/library/74581/asset-preview

Reinforce students' understanding of the discriminant and how it determines the type of solutions for a quadratic equation.

Instructions

1. Provide students with the URL for the interactive game: https://www.media4math.com/library/4833/asset-preview.
2. Instruct students to play the game for 10 minutes. The game requires them to classify quadratic equations based on their discriminants as having two real solutions, one real solution, or two complex solutions.
3. After playing, ask students to write down three equations they classified during the game and their corresponding discriminants.
4. Discuss as a class how the discriminant was calculated for these equations and how it influenced the classification.
5. Transition into a brief review of the quadratic formula, highlighting the role of the discriminant.

Activity 3: Desmos Activity

Use the Desmos graphing calculator to identify the roots of a quadratic equation graphically.

1. Access Desmos:
   • Open the Desmos graphing calculator on your device: Desmos Graphing Calculator.
2. Explore a Quadratic Equation:
   • Enter the quadratic function y = x2 - 4x + 3 into Desmos.
3. Visualize the Graph:
   • Observe the parabola.
4. Identify the Roots:
   • Locate the points where the parabola intersects the x-axis. These are the roots of the equation.
   • Verify that the roots of x2 - 4x + 3 = 0 are x = 1 and x = 3.

5. Try Another Equation:
   • Enter the quadratic equation y = x2 - 2x - 3.
   • Identify the roots graphically and write them down.
   • Discuss with a partner how the equation and graph are related.
6. Reflection Questions:
   • What do the roots of a quadratic equation represent on the graph?
   • How can you tell if a quadratic equation has no real roots just by looking at the graph?

Teach

The quadratic formula is one of the most versatile tools in algebra, providing a systematic method for solving any quadratic equation. It is particularly useful when factoring is not straightforward or when the equation has irrational or complex solutions. Derived from completing the square on the general quadratic equation ax² + bx + c = 0, the quadratic formula is expressed as:

x = (-b ± √(b2 - 4ac)) / 2a

This formula hinges on these key components: the coefficients a, b, and c, and the discriminant, b² - 4ac. The discriminant determines the nature of the solutions: if positive, the equation has two distinct real solutions; if zero, a single real solution exists; and if negative, two complex solutions arise.

In this section, students will explore how to identify the coefficients from a given quadratic equation and substitute them into the quadratic formula. By practicing with a variety of equations, they will gain fluency in applying the formula and interpreting the solutions. This foundational skill reinforces algebraic reasoning and prepares students to tackle more advanced applications, such as graphing parabolas or modeling real-world scenarios. Let’s begin by analyzing the formula step-by-step and solving sample quadratic equations together.

Summary: This section introduces the quadratic formula and the discriminant with three worked examples, including one real-world application.

Refer to the following video tutorials as needed, which walk students through the use of the quadratic formula:

• Video Tutorial: Anatomy of an Equation: Quadratic Equations 1: https://www.media4math.com/library/39961/asset-preview
• Video Tutorial: https://www.media4math.com/library/74288/asset-preview
• Video Tutorial: https://www.media4math.com/library/74289/asset-preview
• Video Tutorial: https://www.media4math.com/library/74290/asset-preview
• Video Tutorial: https://www.media4math.com/library/74291/asset-preview
• Video Tutorial: https://www.media4math.com/library/74292/asset-preview

Example 1

Solve x² + 5x + 6 = 0 using the quadratic formula.

The quadratic formula is:

• Step 1: Identify coefficients: a = 1, b = 5, c = 6.
• Step 2: Calculate the discriminant: 

b² - 4ac = 25 - 24 = 1

This means there are two real roots.

• Step 3: Apply the quadratic formula: 

x = (-5 ± √1)/2
x = (-5 ± 1)/2

• Step 4: 

Simplify: x = -2, -3.

Summary: The equation has two distinct real solutions: x = -2 and x = -3.

Example 2

Solve 2x² + 4x + 2 = 0.

• The quadratic formula is:

• Step 1: Identify coefficients: a = 2, b = 4, c = 2.
• Step 2: Calculate the discriminant: b² - 4ac = 16 - 16 = 0. These means there is only one real root, which also means that the quadratic is a perfect square.
• Step 3: Apply the quadratic formula: x = (-4 ± √0)/4.
• Step 4: Simplify: x = -1.

Summary: The equation has one real solution (a repeated root): x = -1. Notice how this parabola intersects the x-axis just once.

Example 3 (Real World)

A ball is thrown upward from a height of 20 feet with a velocity of 40 ft/s. When will it hit the ground? h(t) = -16t² + 40t + 20.

The quadratic formula is:

• Step 1: Identify coefficients: a = -16, b = 40, c = 20.
• Step 2: Calculate the discriminant: b² - 4ac = 1600 + 1280 = 2880. This means there are two real roots.
• Step 3: Apply the quadratic formula: t = (-40 ± √2880)/(-32).
• Step 4: Simplify: t ≈ -0.42 seconds, 2.9 seconds. Discard the negative root, since time values are positive.

Summary: The ball hits the ground at approximately 2.6 seconds.

Review

Reinforce key points:

• The quadratic formula solves any quadratic equation.
• The discriminant determines the number and type of solutions.

Interactive Activities

Try these drag-and-drop activities to test quadratic equations and the discriminant.

• Interactive Math Game--DragNDrop--Quadratic Equations: https://www.media4math.com/library/4832/asset-preview
• Interactive Math Game--DragNDrop--The Discriminant: https://www.media4math.com/library/4833/asset-preview

Example 1: Solve 2x² - 3x - 2 = 0

1. Identify coefficients: a = 2, b = -3, c = -2

2. Substitute into the formula:

   x = (3 ± √((-3)² - 4(2)(-2))) / (2(2))

3. Simplify inside the square root:

   x = (3 ± √(9 - (-16))) / 4

   x = (3 ± √25) / 4

4. Solve the square root:

   x = (3 ± 5) / 4

5. Split into two solutions:

x₁ = (3 + 5) / 4 = 2

x₂ = (3 - 5) / 4 = -1/2

Final Answer: x = 2 or x = -1/2

Example 2: Solve x² + 6x + 8 = 0

1. Identify coefficients: a = 1, b = 6, c = 8

2. Substitute into the formula:

   x = (-6 ± √(6² - 4(1)(8))) / (2(1))

3. Simplify inside the square root:

   x = (-6 ± √(36 - 32)) / 2

   x = (-6 ± √4) / 2

4. Solve the square root:

   x = (-6 ± 2) / 2

5. Split into two solutions:

x₁ = (-6 + 2) / 2 = -2

x₂ = (-6 - 2) / 2 = -4

Final Answer: x = -2 or x = -4

Example 3: Solve 3x² + x - 4 = 0

1. Identify coefficients: a = 3, b = 1, c = -4

2. Substitute into the formula:

   x = (-1 ± √(1² - 4(3)(-4))) / (2(3))

3. Simplify inside the square root:

   x = (-1 ± √(1 - (-48))) / 6

   x = (-1 ± √49) / 6

4. Solve the square root:

   x = (-1 ± 7) / 6

5. Split into two solutions:

x₁ = (-1 + 7) / 6 = 1

x₂ = (-1 - 7) / 6 = -4/3

Final Answer: x = 1 or x = -4/3

Key Takeaways

• Always identify a, b, and c correctly.
• Perform careful calculations under the square root (discriminant b² - 4ac).
• Simplify results to their simplest form for clarity.

Additional Multimedia Resources

As part of the review include this Video Tutorial: https://www.media4math.com/library/39559/asset-preview

Quiz

Directions: Solve each quadratic equation using the quadratic formula. Show all work.

1. x² - 3x + 2 = 0
   
    
2. 2x² + 4x - 6 = 0
   
    
3. x² + 4x + 5 = 0
   
    
4. 3x² - 2x - 1 = 0
   
    
5. x² - x - 12 = 0
   
    
6. 4x² + 8x + 4 = 0
   
    
7. x² + 6x + 9 = 0
   
    
8. x² + 2x - 8 = 0
   
    
9. 5x² - 10x + 5 = 0
   
    
10. x² - 2x - 15 = 0

Answer Key

1. x = 1, 2
2. x = 1, -3
3. No real solutions
4. x = 1, -1/3
5. x = 4, -3
6. x = -1 (repeated root)
7. x = -3 (repeated root)
8. x = 2, -4
9. x = 1 (repeated root)
10. x = 5, -3