Mastering Completing the Square: A Comprehensive Guide

Algebra 2 Grades High School 5:36 Video
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Lesson Description

Learn how to solve quadratic equations by completing the square with step-by-step examples and practice problems. This lesson covers finding the 'c' value, factoring, and solving equations.

Video Resource

Completing the Square

Mario's Math Tutoring

Duration: 5:36
Watch on YouTube

Key Concepts

  • Perfect Square Trinomials
  • Finding the 'c' value to complete the square
  • Solving quadratic equations by completing the square

Learning Objectives

  • Students will be able to identify and create perfect square trinomials.
  • Students will be able to solve quadratic equations by completing the square.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing quadratic equations and the concept of perfect square trinomials. Briefly explain why completing the square is a useful technique for solving certain types of quadratic equations.
  • Finding the 'c' Value (10 mins)
    Watch the video from 0:18 to 2:06. Guide students through the process of finding the 'c' value to make a trinomial a perfect square. Emphasize the steps: take half of the coefficient of the x term, square it, and that's the 'c' value.
  • Solving Equations by Completing the Square (20 mins)
    Watch the video from 2:27 to 4:08. Demonstrate how to solve quadratic equations by completing the square. Stress the importance of adding the same value to both sides of the equation to maintain balance. Show how to take the square root of both sides and solve for x, remembering the plus or minus.
  • Challenging Example (15 mins)
    Watch the video from 4:08 to the end. Explain how to complete the square when the coefficient of x^2 is not 1. Walk them through dividing all terms by the coefficient of x^2 before completing the square.
  • Practice and Review (15 mins)
    Provide students with practice problems to solve independently or in small groups. Circulate and provide assistance as needed. Review the key steps and address any remaining questions.

Interactive Exercises

  • Perfect Square Identification
    Present students with a series of trinomials and ask them to identify which ones are perfect square trinomials. If a trinomial is not a perfect square, have them find the 'c' value that would make it a perfect square.
  • Completing the Square Practice
    Give students a set of quadratic equations to solve by completing the square. Start with simpler equations and gradually increase the difficulty.

Discussion Questions

  • Why is it important to add the same value to both sides of the equation when completing the square?
  • How does completing the square relate to factoring perfect square trinomials?
  • What are the advantages and disadvantages of using completing the square compared to other methods for solving quadratic equations, such as factoring or the quadratic formula?

Skills Developed

  • Algebraic manipulation
  • Problem-solving
  • Critical thinking

Multiple Choice Questions

Question 1:

To complete the square for the expression x^2 + 8x + c, what value of 'c' is needed?

Correct Answer: 16

Question 2:

When solving a quadratic equation by completing the square, why do we add the same value to both sides of the equation?

Correct Answer: Both B and C

Question 3:

What is the factored form of the perfect square trinomial x^2 - 6x + 9?

Correct Answer: (x - 3)^2

Question 4:

When taking the square root of both sides of an equation, it's important to remember:

Correct Answer: Both the positive and negative roots

Question 5:

Before completing the square for the equation 2x^2 + 4x - 6 = 0, what should you do?

Correct Answer: Divide all terms by 2

Question 6:

What is half of the middle coefficient in the expression x^2 + 7x + c?

Correct Answer: 3.5

Question 7:

What is the value of c that completes the square for x^2 - 5x + c ?

Correct Answer: 6.25

Question 8:

When solving by completing the square, after you isolate the perfect square and take the square root, what must you remember to include?

Correct Answer: Both positive and negative solutions

Question 9:

What is the first step in completing the square for the equation x^2 + 6x -5 = 0?

Correct Answer: Move the constant term to the right side

Question 10:

Which of the following equations is set up correctly to solve by completing the square?

Correct Answer: All of the above

Fill in the Blank Questions

Question 1:

To find the 'c' value to complete the square, you take half of the coefficient of the x term and then __________ it.

Correct Answer: square

Question 2:

Completing the square is a method used to solve __________ equations.

Correct Answer: quadratic

Question 3:

A perfect square trinomial can be factored into the form (x + a)^2 or (x - a)^2, where 'a' is half of the coefficient of the __________ term.

Correct Answer: x

Question 4:

When completing the square, it is crucial to add the same value to __________ sides of the equation to maintain balance.

Correct Answer: both

Question 5:

Before completing the square, if the coefficient of x^2 is not 1, you must __________ all terms by that coefficient.

Correct Answer: divide

Question 6:

The constant term that completes x^2 + 12x + ____ is 36.

Correct Answer: 36

Question 7:

When solving by completing the square, the step after taking the square root of both sides is to __________ x.

Correct Answer: isolate

Question 8:

If an equation is (x + 5)^2 = 16, the two solutions are obtained by computing -5 + 4 and -5 - ____

Correct Answer: 4

Question 9:

When completing the square, remember to move the __________ to the right side of the equation initially.

Correct Answer: constant

Question 10:

If you have x^2 + 8x = 10, you need to add ______ to both sides in order to complete the square.

Correct Answer: 16

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.2.1:Factor polynomial expressions including, but not limited to, trinomials, differences of squares, sum and difference of cubes, and factoring by grouping, using a variety of tools and strategies.

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