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Algebra 2
Mastering Completing the Square: Solving Quadratic Equations with Non-Unit Leading Coefficients

Mastering Completing the Square: Solving Quadratic Equations with Non-Unit Leading Coefficients

Algebra 2 Grades High School 2:19 Video

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Lesson Description

Learn how to solve quadratic equations by completing the square, focusing on cases where the leading coefficient is not equal to 1. This lesson provides a step-by-step approach suitable for Algebra 2 students.

Video Resource

Solving a Quadratic Equation by Completing the Square

Mario's Math Tutoring

Duration: 2:19

Watch on YouTube

Key Concepts

Quadratic Equations
Completing the Square
Leading Coefficient
Perfect Square Trinomials
Square Root Property

Learning Objectives

Students will be able to solve quadratic equations by completing the square when the leading coefficient is not 1.
Students will be able to manipulate quadratic equations to create perfect square trinomials.
Students will be able to apply the square root property to solve for the variable.

Educator Instructions

Introduction (5 mins)
Begin by reviewing the standard form of a quadratic equation and the concept of solving equations by isolating the variable. Briefly discuss why completing the square is a useful technique, especially when factoring is not straightforward. Show an example of a perfect square trinomial.
Video Demonstration (10 mins)
Play the Mario's Math Tutoring video: 'Solving a Quadratic Equation by Completing the Square'. Encourage students to take notes on the steps demonstrated. Emphasize the importance of dividing by the leading coefficient as the initial step.
Step-by-Step Breakdown (15 mins)
Go through the steps outlined in the video, pausing to explain each one in detail: 1. **Divide by the leading coefficient:** Make the coefficient of the x² term equal to 1. 2. **Isolate the constant term:** Move the constant term to the right side of the equation. 3. **Complete the square:** Take half of the coefficient of the x term, square it, and add it to both sides of the equation. 4. **Factor the perfect square trinomial:** Rewrite the left side as a squared binomial. 5. **Apply the square root property:** Take the square root of both sides, remembering to include both positive and negative roots. 6. **Solve for x:** Isolate x to find the two solutions.
Guided Practice (15 mins)
Work through an example problem together as a class, guiding students through each step. Ask questions to ensure understanding at each stage.
Independent Practice (15 mins)
Provide students with practice problems to solve independently. Circulate the classroom to offer assistance and answer questions.
Review and Wrap-up (5 mins)
Review the key steps of completing the square. Address any remaining questions and provide a brief overview of when this method is most useful.

Interactive Exercises

Error Analysis
Provide students with worked-out examples containing common errors. Ask them to identify and correct the mistakes.
Group Problem Solving
Divide students into small groups and assign each group a different quadratic equation to solve by completing the square. Have each group present their solution to the class.

Discussion Questions

Why is it important to have a leading coefficient of 1 when completing the square?
How does completing the square relate to writing a quadratic function in vertex form?
What are the advantages and disadvantages of completing the square compared to other methods for solving quadratic equations (e.g., factoring, 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 when the leading coefficient is not 1?

Isolate the constant term.

Divide every term by the leading coefficient.

Take the square root of both sides.

Factor the quadratic expression.

Correct Answer: Divide every term by the leading coefficient.

Question 2:

When completing the square, what value do you add to both sides of the equation?

Half the coefficient of the x term.

The square of half the coefficient of the x term.

The coefficient of the x² term.

The square root of the constant term.

Correct Answer: The square of half the coefficient of the x term.

Question 3:

What is the purpose of 'completing the square'?

To eliminate the x term.

To create a perfect square trinomial.

To make the equation easier to factor using difference of squares.

To isolate the variable x.

Correct Answer: To make the equation easier to factor using difference of squares.

Question 4:

After completing the square and factoring, what is the next step to solve for x?

Add the constant term to both sides.

Take the square root of both sides.

Divide both sides by 2.

Multiply both sides by -1.

Correct Answer: Take the square root of both sides.

Question 5:

Why do we include a 'plus or minus' sign when taking the square root of both sides of an equation?

Because squaring a positive or negative number results in a positive number.

Because the square root of a negative number is imaginary.

To ensure we find all possible solutions.

All of the above.

Correct Answer: All of the above.

Question 6:

Solve for x by completing the square: x² - 4x = 5

x = 5, x = -1

x = 1, x = -5

x = 9, x = -1

x = 3, x = -3

Correct Answer: x = 5, x = -1

Question 7:

Solve for x by completing the square: 2x² + 8x - 10 = 0

x = 1, x = -5

x = -1, x = 5

x = 2, x = -5

x = -2, x = 5

Correct Answer: x = 1, x = -5

Question 8:

What type of expression is created on one side of the equation after completing the square?

A linear expression

A perfect square trinomial

A rational expression

An exponential expression

Correct Answer: A perfect square trinomial

Question 9:

Which of the following equations is set up correctly to complete the square?

2x² + 4x + 8 = 0

x² - 6x = 20

x² + 5x + 2 = 0

3x² = 9x - 3

Correct Answer: x² - 6x = 20

Question 10:

Given the equation (x + 3)² = 16, what are the solutions for x?

x = 4, x = -4

x = 1, x = -7

x = 7, x = -1

x = 1, x = -1

Correct Answer: x = 1, x = -7

Fill in the Blank Questions

Question 1:

Before completing the square, if the leading coefficient is not 1, you must _____ every term by the leading coefficient.

Correct Answer: divide

Question 2:

To complete the square, you add the _____ of half the coefficient of the x term to both sides of the equation.

Correct Answer: square

Question 3:

Completing the square transforms a quadratic expression into a _____ _____ _____, which can be factored as a binomial squared.

Correct Answer: perfect square trinomial

Question 4:

When taking the square root of both sides of an equation, remember to include both the _____ and _____ roots.

Correct Answer: positive and negative

Question 5:

The solutions to a quadratic equation are also known as the _____ or _____ of the equation.

Correct Answer: roots and zeros

Question 6:

In the equation x² + 6x + ____ = (x + 3)², the missing term is ____

Correct Answer: 9

Question 7:

To solve (x - 5)² = 9, the first step after completing the square is to take the _____ _____ of both sides.

Correct Answer: square root

Question 8:

After dividing by the leading coefficient and moving the constant, the resulting equation is x² - 8x = 12. To complete the square, you add ____ to both sides.

Correct Answer: 16

Question 9:

If the final form of the equation after completing the square is (x + 2)² = 5, then x = ____ or x = ____.

Correct Answer: -2 + √5 or -2 - √5

Question 10:

The method of completing the square is useful when a quadratic equation cannot be easily _____.

Correct Answer: factored

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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