Mastering Completing the Square: Advanced Techniques

Algebra 2 Grades High School 5:21 Video
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Lesson Description

Learn how to solve quadratic equations by completing the square when the leading coefficient is not equal to 1. This lesson covers factoring, balancing equations, and writing equations in vertex form.

Video Resource

How to Complete the Square when a≠1

Mario's Math Tutoring

Duration: 5:21
Watch on YouTube

Key Concepts

  • Factoring out leading coefficients
  • Balancing equations when completing the square
  • Perfect square trinomials
  • Vertex form of a quadratic equation

Learning Objectives

  • Students will be able to complete the square when the leading coefficient is not 1.
  • Students will be able to solve quadratic equations with non-unity leading coefficients by completing the square.
  • Students will be able to convert quadratic equations to vertex form by completing the square.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the basic steps of completing the square when the leading coefficient is 1. Briefly discuss why the process is slightly different when the leading coefficient is not 1.
  • Example 1: Solving a Quadratic Equation (15 mins)
    Play the first example from the video (0:16-3:09). Pause at each step to explain the process in detail. Emphasize the importance of factoring out the leading coefficient correctly and balancing the equation. Walk through the following steps: 1. Move the constant term to the right side of the equation. 2. Factor out the leading coefficient from the left side. 3. Take half of the coefficient of the x-term inside the parentheses, square it, and add it inside the parentheses. 4. Balance the equation by adding the appropriate value to the right side (remember to account for the factored-out coefficient). 5. Factor the perfect square trinomial. 6. Solve for x by isolating the squared term, taking the square root of both sides (remembering both positive and negative roots), and isolating x.
  • Example 2: Converting to Vertex Form (10 mins)
    Play the second example from the video (3:09 onwards). Explain how completing the square can be used to convert a quadratic equation to vertex form. Stress that the process is very similar to solving an equation, but the goal is different (to rewrite the equation). Follow these steps: 1. Factor out the leading coefficient from the $x^2$ and $x$ terms. 2. Complete the square inside the parentheses. Remember to add/subtract appropriately on the same side of the equation to balance. 3. Rewrite the trinomial as a perfect square and simplify. 4. You should now have the equation in vertex form: $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex.
  • Practice Problems (15 mins)
    Provide students with practice problems that require them to complete the square with non-unity leading coefficients, both for solving equations and converting to vertex form. Encourage them to work independently or in pairs and provide assistance as needed.
  • Wrap-up (5 mins)
    Review the key steps of completing the square with non-unity leading coefficients. Answer any remaining questions and emphasize the connection between completing the square, solving quadratic equations, and converting to vertex form.

Interactive Exercises

  • Error Analysis
    Present students with worked-out examples of completing the square that contain common errors. Have them identify the errors and correct them.
  • Group Challenge
    Divide students into groups and give each group a challenging problem involving completing the square (e.g., a word problem that requires them to set up and solve a quadratic equation). The first group to solve the problem correctly wins.

Discussion Questions

  • Why is it necessary to factor out the leading coefficient before completing the square?
  • What happens if you don't balance the equation correctly when completing the square?
  • How does completing the square help you find the vertex of a quadratic function?
  • Can completing the square be used when the quadratic equation has imaginary roots?

Skills Developed

  • Algebraic manipulation
  • Problem-solving
  • Critical thinking
  • Attention to detail

Multiple Choice Questions

Question 1:

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

Correct Answer: Factor out the leading coefficient from the x² and x terms.

Question 2:

When factoring out a leading coefficient of 1/3 from the expression (1/3)x² - 6x, what is the resulting x term inside the parentheses?

Correct Answer: -18x

Question 3:

While completing the square, you factor out a '4' from 4x² + 8x. You then add 1 inside the parentheses to complete the square. What value must you add to the other side of the equation to balance it?

Correct Answer: 4

Question 4:

After completing the square, you have (x + 3)² = 16. What are the solutions for x?

Correct Answer: x = -7, 1

Question 5:

The vertex form of a quadratic equation is y = a(x - h)² + k. What does the (h, k) represent?

Correct Answer: The vertex of the parabola

Question 6:

If you have (1/2)(x - 4)² = 8, what should you do to begin isolating the squared term?

Correct Answer: Multiply by 2

Question 7:

Why is it important to consider both positive and negative roots when solving by taking the square root?

Correct Answer: Both A and B

Question 8:

What is the value that completes the square inside the parenthesis for the expression: x² -10x + ____?

Correct Answer: 25

Question 9:

What is vertex form?

Correct Answer: y = a(x - h)² + k

Question 10:

Which of the following equations shows the correct first step of completing the square for: 2x² + 8x - 10 = 0?

Correct Answer: 2x² + 8x = 10

Fill in the Blank Questions

Question 1:

Before completing the square, if 'a' ≠ 1, you must ______ out the leading coefficient from the x² and x terms.

Correct Answer: factor

Question 2:

When solving by completing the square, after isolating the squared term, you take the ______ ______ of both sides.

Correct Answer: square root

Question 3:

The value you add to complete the square is found by taking half of the coefficient of the x term and ______ it.

Correct Answer: squaring

Question 4:

To balance the equation, when you add a value inside parentheses that are being multiplied by a coefficient, you must multiply that value by the ______ before adding it to the other side.

Correct Answer: coefficient

Question 5:

The vertex form of a quadratic equation is y = a(x - h)² + k, where (h, k) represents the ______.

Correct Answer: vertex

Question 6:

Completing the square transforms a quadratic equation into ______ form.

Correct Answer: vertex

Question 7:

When taking the square root of both sides, it is important to consider both positive and ______ roots.

Correct Answer: negative

Question 8:

If you are factoring out 2 from 2x² + 6x, the resulting expression inside the parenthesis will be x² + ______.

Correct Answer: 3x

Question 9:

In the equation (x + 5)² = 9, the next step to solve for x is to take the square root of ______.

Correct Answer: 9

Question 10:

To counterbalance dividing by 3, you would multiple by ______.

Correct Answer: 3

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. A2.F.1.3:Graph a quadratic function. Identify the domain, range, x- and y-intercepts, maximum or minimum value, axis of symmetry, and vertex using various methods and tools that may include a graphing calculator or appropriate technology.

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