Mastering Quadratics: Completing the Square Demystified

Algebra 2 Grades High School 6:20 Video
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Lesson Description

Learn to solve quadratic equations by completing the square with a step-by-step approach. This lesson covers equations with leading coefficients of 1 and greater than 1.

Video Resource

Solve Quadratic Equations by Completing the Square (6 Easy Steps)

Mario's Math Tutoring

Duration: 6:20
Watch on YouTube

Key Concepts

  • Quadratic Equations
  • Completing the Square
  • Perfect Square Trinomials
  • Leading Coefficient

Learning Objectives

  • Students will be able to solve quadratic equations with a leading coefficient of 1 by completing the square.
  • Students will be able to solve quadratic equations with a leading coefficient not equal to 1 by completing the square.
  • Students will be able to identify and create perfect square trinomials.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the standard form of a quadratic equation (ax² + bx + c = 0). Briefly discuss why completing the square is a useful technique for solving quadratic equations, especially when factoring is not straightforward. Introduce the video by Mario's Math Tutoring and explain that it will demonstrate a step-by-step method for completing the square.
  • Video Viewing and Note-Taking (15 mins)
    Play the video 'Solve Quadratic Equations by Completing the Square (6 Easy Steps)' by Mario's Math Tutoring. Instruct students to take detailed notes on each of the 6 steps, including the examples provided in the video. Encourage students to pause the video as needed to fully understand each step. Emphasize writing down the steps for both types of problems, where a=1 and where a!=1.
  • Guided Practice (20 mins)
    Work through additional examples on the board, reinforcing the steps outlined in the video. Start with simpler examples (a=1) and gradually progress to more complex problems (a!=1). Encourage student participation by asking them to identify the 'a', 'b', and 'c' values and guide you through each step. Address any questions or misconceptions that arise.
  • Independent Practice (15 mins)
    Assign practice problems for students to solve independently. Circulate the classroom to provide assistance and monitor progress. Offer hints and guidance as needed, but encourage students to rely on their notes and the video's examples.
  • Wrap-up and Assessment (5 mins)
    Summarize the key steps involved in completing the square. Answer any remaining questions. Administer a short multiple-choice quiz to assess student understanding (see below).

Interactive Exercises

  • Error Analysis
    Present students with worked-out problems that contain common errors in completing the square. Have them identify the errors and correct them.
  • Group Problem Solving
    Divide the class into small groups and assign each group a challenging quadratic equation to solve by completing the square. Have each group present their solution to the class.

Discussion Questions

  • Why is it important to divide by the leading coefficient before completing the square?
  • How does completing the square relate to the quadratic formula?
  • Can completing the square be used to solve any quadratic equation? Explain.

Skills Developed

  • Algebraic Manipulation
  • Problem-Solving
  • Critical Thinking
  • Attention to Detail

Multiple Choice Questions

Question 1:

What is the first step in completing the square?

Correct Answer: Move the constant term to the other side of the equation.

Question 2:

What do you need to do if the leading coefficient is not 1?

Correct Answer: Divide the entire equation by the leading coefficient.

Question 3:

To complete the square, you add (b/2)² to both sides of the equation. What does 'b' represent?

Correct Answer: The coefficient of the x term.

Question 4:

When taking the square root of both sides of an equation, what must you remember?

Correct Answer: Both the positive and negative roots.

Question 5:

What type of trinomial are we trying to create on one side of the equation?

Correct Answer: Perfect Square Trinomial

Question 6:

In the equation x² + 6x + 5 = 0, what is the value of (b/2)²?

Correct Answer: 9

Question 7:

After completing the square, how do you factor the perfect square trinomial x² - 8x + 16?

Correct Answer: (x - 4)²

Question 8:

Which equation is set up correctly to complete the square for 2x² - 8x + 6 = 0?

Correct Answer: x² - 4x = -3

Question 9:

What are the solutions to the equation (x - 3)² = 4?

Correct Answer: x = 5, x = 1

Question 10:

What is the value of 'a' called?

Correct Answer: The leading coefficient

Fill in the Blank Questions

Question 1:

The process of rewriting a quadratic equation to easily solve for the roots is called completing the ______.

Correct Answer: square

Question 2:

Before completing the square, if the leading coefficient is not 1, you must _______ the entire equation by that coefficient.

Correct Answer: divide

Question 3:

To find the value to add to both sides, you take one-half of the ______ value and square it.

Correct Answer: b

Question 4:

When you take the square root of both sides of an equation, you must include both the positive and _______ roots.

Correct Answer: negative

Question 5:

A trinomial that can be factored into (ax + b)² or (ax - b)² is called a _______ _______ trinomial.

Correct Answer: perfect square

Question 6:

In the quadratic equation ax² + bx + c = 0, 'c' represents the _______ term.

Correct Answer: constant

Question 7:

The factored form after completing the square will always be a binomial _______.

Correct Answer: squared

Question 8:

The opposite of squaring is taking the _______ _______.

Correct Answer: square root

Question 9:

The number in front of the squared variable is called the ______ ______.

Correct Answer: leading coefficient

Question 10:

The constants a, b, and c are always _______ numbers.

Correct Answer: real

Educational Standards

A2.A.1.1:Use mathematical models to represent quadratic relationships and solve using factoring, completing the square, the quadratic formula, and various methods (including graphing calculator or other appropriate technology). Find non-real roots when they exist. A2.A.2:Generate and evaluate equivalent algebraic expressions and equations using various strategies.

Teaching Materials

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