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?

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?

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

Question 3:

What is the purpose of 'completing the square'?

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?

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?

Correct Answer: All of the above.

Question 6:

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

Correct Answer: x = 5, x = -1

Question 7:

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

Correct Answer: x = 1, x = -5

Question 8:

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

Correct Answer: A perfect square trinomial

Question 9:

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

Correct Answer: x² - 6x = 20

Question 10:

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

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.

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