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Algebra 2
Mastering Completing the Square: A Comprehensive Guide

Mastering Completing the Square: A Comprehensive Guide

Algebra 2 Grades High School 21:08 Video

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Lesson Description

Learn to solve quadratic equations by completing the square. This lesson covers the fundamental concepts and step-by-step methods, with examples ranging from basic to complex, including those with fractions.

Video Resource

Solve Equations Completing the Square

Kevinmathscience

Duration: 21:08

Watch on YouTube

Key Concepts

Quadratic Equations
Completing the Square Method
Square Roots and Plus/Minus Solutions
Rearranging Equations

Learning Objectives

Students will be able to rewrite quadratic equations in completed square form.
Students will be able to solve quadratic equations by using the method of completing the square, including those with fractional coefficients.
Students will be able to identify and express solutions in radical form, simplified radical form, and decimal approximations.

Educator Instructions

Introduction (5 mins)
Begin by briefly reviewing quadratic equations and traditional methods for solving them (factoring, quadratic formula). Introduce the concept of completing the square as an alternative method that is especially useful when factoring is difficult.
Video Viewing and Note-Taking (20 mins)
Play the 'Solve Equations Completing the Square' video by Kevinmathscience. Instruct students to take detailed notes, focusing on the steps involved in completing the square and the examples provided. Encourage them to pause the video and re-watch sections as needed.
Guided Practice (20 mins)
Work through several example problems on the board, demonstrating each step of the completing the square method. Start with simpler examples and gradually increase complexity, including examples with fractional coefficients. Have students follow along in their notebooks.
Independent Practice (20 mins)
Provide students with a set of practice problems to solve independently. Circulate the room to provide assistance and answer questions. Encourage students to work together and discuss their solutions.
Review and Assessment (15 mins)
Review the solutions to the practice problems as a class. Address any remaining questions or misconceptions. Administer a short quiz to assess student understanding of the completing the square method.

Interactive Exercises

Completing the Square Challenge
Divide the class into groups and assign each group a different quadratic equation to solve by completing the square. Have each group present their solution to the class, explaining each step of their process.
Error Analysis
Provide students with examples of completed problems that contain common errors in the completing the square process. Have students identify the errors and correct them.

Discussion Questions

Why is it important to have a coefficient of 1 in front of the x² term before completing the square?
When taking the square root of both sides of an equation, why do we need to consider both positive and negative solutions?
In what situations might completing the square be a more efficient method for solving quadratic equations than factoring or using the quadratic formula?

Skills Developed

Algebraic Manipulation
Problem-Solving
Critical Thinking
Attention to detail

Multiple Choice Questions

Question 1:

What is the first step in solving a quadratic equation by completing the square if the coefficient of the \(x^2\) term is not 1?

Divide every term by the coefficient of the \(x^2\) term.

Move the constant term to the right side of the equation.

Take the square root of both sides of the equation.

Add a constant to both sides to make the left side a perfect square trinomial.

Correct Answer: Divide every term by the coefficient of the \(x^2\) term.

Question 2:

To complete the square for the expression \(x^2 + bx\), what constant must be added?

\(b\)

\(b^2\)

\((b/2)^2\)

\(b/2\)

Correct Answer: \((b/2)^2\)

Question 3:

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

Only the positive root.

Only the negative root.

Both the positive and negative roots.

The imaginary unit \(i\).

Correct Answer: Both the positive and negative roots.

Question 4:

Solve for x: \(x^2 + 6x = 7\)

x = -7, x = 1

x = 7, x = -1

x = 1, x = -1

x = 4, x = -4

Correct Answer: x = -7, x = 1

Question 5:

What is the completed square form of \(x^2 - 8x + 16\)?

\((x + 4)^2\)

\((x - 4)^2\)

\((x + 8)^2\)

\((x - 8)^2\)

Correct Answer: \((x - 4)^2\)

Question 6:

Solve for x: \(x^2 + 4x - 5 = 0\)

x = -1, x = 5

x = 1, x = -5

x = 2, x = -3

x = -2, x = 3

Correct Answer: x = 1, x = -5

Question 7:

What is the value of 'c' that completes the square for \(x^2 - 10x + c\)?

-25

-10

Correct Answer: 25

Question 8:

Rewrite the following equation in completed square form: \(x^2 + 8x + 12 = 0\)

\((x + 4)^2 = 4\)

\((x - 4)^2 = 4\)

\((x + 2)^2 = 16\)

\((x - 2)^2 = 16\)

Correct Answer: \((x + 4)^2 = 4\)

Question 9:

Solve for x: \(2x^2 - 8x - 10 = 0\)

x = 5, x = -1

x = -5, x = 1

x = 1, x = -1

x = 4, x = -4

Correct Answer: x = 5, x = -1

Question 10:

Solve for x using completing the square: \(x^2 - 2x - 1 = 0\)

x = 1 ± √2

x = -1 ± √2

x = 1 ± √3

x = -1 ± √3

Correct Answer: x = 1 ± √2

Fill in the Blank Questions

Question 1:

Before completing the square, the coefficient of the \(x^2\) term must be ____.

Correct Answer: 1

Question 2:

To complete the square for \(x^2 - 12x\), you need to add the constant ____.

Correct Answer: 36

Question 3:

When solving by completing the square, remember to include both the positive and ____ square root.

Correct Answer: negative

Question 4:

The process of completing the square transforms a quadratic equation into the form \((x + a)^2 = ____\).

Correct Answer: k

Question 5:

When completing the square, you take half of the coefficient of the x term, ____ it, and add it to both sides of the equation.

Correct Answer: square

Question 6:

To solve \(x^2 + 10x = -9\) by completing the square, first add ____ to both sides.

Correct Answer: 25

Question 7:

The completed square form of \(x^2 - 6x + 5 = 0\) is \((x - 3)^2 = ____\).

Correct Answer: 4

Question 8:

To complete the square for the equation \(x^2 + 5x + c\), the value of c must be ____.

Correct Answer: 25/4

Question 9:

When solving the equation \(x^2 + 4x = 21\) by completing the square, the values for x are 3 and ____.

Correct Answer: -7

Question 10:

In completing the square, if the original equation has a constant term on the same side as the x terms, the first step is to ____ it to the other side.

Correct Answer: move

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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