Mastering Completing the Square: A Quadratic Equation Solver's Toolkit

Algebra 2 Grades High School 7:15 Video
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Lesson Description

Learn how to solve quadratic equations by completing the square with increasing difficulty, covering various scenarios including leading coefficients and fractions.

Video Resource

Completing the Square to Solve a Quadratic Equation (3 Examples)

Mario's Math Tutoring

Duration: 7:15
Watch on YouTube

Key Concepts

  • Quadratic Equations
  • Completing the Square
  • Perfect Square Trinomials

Learning Objectives

  • Students will be able to solve quadratic equations by completing the square.
  • Students will be able to manipulate quadratic equations to prepare them for completing the square.
  • Students will be able to handle quadratic equations with leading coefficients and fractional coefficients.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the standard form of a quadratic equation and the importance of solving them. Briefly introduce the concept of completing the square as an alternative method to factoring or using the quadratic formula. Show a simple quadratic equation like x^2 + 4x + 3 = 0 and ask students how they would solve it.
  • Example 1: Basic Completing the Square (10 mins)
    Follow the video's first example (x^2 + 6x - 7 = 0). Emphasize the steps: isolating the variables, taking half of the x-coefficient, squaring it, and adding it to both sides. Explain why adding the same value to both sides maintains equality. Show how the left side factors into a perfect square trinomial. Conclude by taking the square root and solving for x.
  • Example 2: Leading Coefficient (15 mins)
    Work through the second example (5x^2 - 10x - 15 = 0). Stress the importance of dividing by the leading coefficient before completing the square. Discuss why this step is necessary. Reiterate the steps of completing the square with the modified equation. Introduce the alternative method of adding one to both sides of the equation.
  • Example 3: Fractions (15 mins)
    Tackle the third example (-2x^2 + 5x - 1 = 0). Reinforce dividing by the leading coefficient (which is negative in this case). Carefully demonstrate handling fractions when taking half of the x-coefficient and squaring it. Show how to get a common denominator when adding to both sides. Explain how to simplify the solution with a square root.
  • Practice Problems (15 mins)
    Provide students with practice problems of varying difficulty. Encourage them to work individually or in pairs. Circulate to provide assistance and answer questions.
  • Wrap-up (5 mins)
    Review the key steps of completing the square. Summarize the types of problems covered (basic, leading coefficient, fractions). Answer any remaining questions.

Interactive Exercises

  • Error Analysis
    Present students with a worked-out problem where an error was made during the completing the square process. Have them identify and correct the error.
  • Group Challenge
    Divide the class into groups. Give each group a complex quadratic equation and have them race to solve it by completing the square. The first group to correctly solve the problem wins.

Discussion Questions

  • Why do we need to divide by the leading coefficient before completing the square?
  • What are the advantages and disadvantages of completing the square compared to using the quadratic formula?
  • How does completing the square relate to the vertex form of a quadratic equation?

Skills Developed

  • Algebraic manipulation
  • Problem-solving
  • Critical thinking

Multiple Choice Questions

Question 1:

What is the first step in completing the square when the leading coefficient is not 1?

Correct Answer: Divide every term by the leading coefficient.

Question 2:

To complete the square for x^2 + bx, what do you add to both sides of the equation?

Correct Answer: (b/2)^2

Question 3:

When completing the square, what does the expression x^2 - 8x + 16 factor into?

Correct Answer: (x - 4)^2

Question 4:

After completing the square and taking the square root of both sides, what must you remember to include?

Correct Answer: Both positive and negative roots.

Question 5:

In the equation x^2 + 10x - 3 = 0, what value should be added to both sides to complete the square?

Correct Answer: 25

Question 6:

What is the solution to the equation (x + 2)^2 = 9?

Correct Answer: x = 1 and x = -5

Question 7:

If you have 2x^2 + 8x - 10 = 0, what is the first step to solve this equation by completing the square?

Correct Answer: Divide every term by 2

Question 8:

After completing the square, you have (x - 3)^2 = 7. What is the next step to solving for x?

Correct Answer: Take the square root of both sides

Question 9:

Which of the following equations is already in completed square form?

Correct Answer: (x - 5)^2 = 16

Question 10:

When completing the square with fractions, it's important to:

Correct Answer: Find a common denominator

Fill in the Blank Questions

Question 1:

The process of rewriting a quadratic equation to create a perfect square trinomial is called completing the ________.

Correct Answer: square

Question 2:

To complete the square for x^2 + bx, you add (b/2)^2 to both sides. This value is half of the ________ coefficient squared.

Correct Answer: x

Question 3:

If a quadratic equation has a ________ coefficient other than 1, you must divide every term by this coefficient before completing the square.

Correct Answer: leading

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:

The expression on one side of the equation after completing the square always factors into a ________ squared.

Correct Answer: binomial

Question 6:

Before completing the square, isolate the constant term on the ________ side of the equation.

Correct Answer: right

Question 7:

If the equation is 3x^2 + 6x - 9 = 0, the first step is to divide all the terms by ________.

Correct Answer: 3

Question 8:

When completing the square, (x + a)^2 expands to x^2 + 2ax + ________.

Correct Answer: a^2

Question 9:

Completing the square can be used to rewrite a quadratic equation in ________ form.

Correct Answer: vertex

Question 10:

After completing the square, solve for 'x' by isolating the squared term and then taking the ________ of both sides.

Correct Answer: square root

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.2.2:Add, subtract, multiply, divide, and simplify polynomial expressions.

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