Unlocking Vertex Form: Mastering Completing the Square

Algebra 2 Grades High School 6:10 Video
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Lesson Description

Learn how to rewrite quadratic equations from standard form to vertex form by completing the square. This lesson covers examples where the leading coefficient is both equal to and not equal to one.

Video Resource

Write a Quadratic Equation in Vertex Form by Completing the Square (2 Examples)

Mario's Math Tutoring

Duration: 6:10
Watch on YouTube

Key Concepts

  • Standard Form of a Quadratic Equation (y = ax² + bx + c)
  • Vertex Form of a Quadratic Equation (y = a(x - h)² + k)
  • Completing the Square Technique
  • Vertex of a Parabola

Learning Objectives

  • Students will be able to rewrite a quadratic equation from standard form to vertex form by completing the square.
  • Students will be able to identify the vertex of a parabola given its equation in vertex form.
  • Students will be able to apply the completing the square technique to quadratic equations where the leading coefficient (a) is equal to 1.
  • Students will be able to apply the completing the square technique to quadratic equations where the leading coefficient (a) is not equal to 1.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the standard form and vertex form of a quadratic equation. Briefly discuss the advantages of vertex form (identifying the vertex easily). Introduce the concept of 'completing the square' as a method to transform between these forms.
  • Example 1: Completing the Square (a = 1) (10 mins)
    Work through the first example from the video (y = x² - 14x + 1). Follow the steps outlined in the video: subtract the constant term, take (b/2)², add it to both sides, factor the perfect square trinomial, and isolate y. Emphasize each step and the reasoning behind it.
  • Example 2: Completing the Square (a ≠ 1) (15 mins)
    Work through the second example from the video (f(x) = 5x² - 40x + 7). Highlight the additional step of factoring out the leading coefficient (a). Stress the importance of multiplying the added constant by the factored coefficient when balancing the equation. Explain why you don't want to factor out the greatest common factor, only the leading coefficient.
  • Practice Problems (15 mins)
    Provide students with practice problems of varying difficulty (some with a = 1, some with a ≠ 1). Circulate and provide assistance as needed. Encourage students to work together and discuss their solutions.
  • Wrap-up (5 mins)
    Review the key steps of completing the square. Reiterate the importance of careful balancing and attention to the leading coefficient. Answer any remaining questions.

Interactive Exercises

  • Group Problem Solving
    Divide students into groups and assign each group a different quadratic equation in standard form. Have them work together to rewrite the equation in vertex form and identify the vertex. Each group presents their solution to the class.
  • Error Analysis
    Present students with a worked-out example of completing the square that contains a common error (e.g., forgetting to multiply the added constant by the leading coefficient). Have students identify the error and correct it.

Discussion Questions

  • Why is it useful to rewrite a quadratic equation in vertex form?
  • What happens to the graph of a quadratic function when it's in vertex form, and the 'h' value changes?
  • What is the most challenging part of completing the square, and how can you overcome it?
  • How does vertex form help you to identify the maximum or minimum value of a quadratic function?

Skills Developed

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

Multiple Choice Questions

Question 1:

What is the vertex form of a quadratic equation?

Correct Answer: y = a(x - h)² + k

Question 2:

When completing the square, what do you do with the 'b' value?

Correct Answer: Divide it by 2 and square it

Question 3:

In vertex form y = a(x - h)² + k, what does (h, k) represent?

Correct Answer: The vertex

Question 4:

When completing the square, if you add a value to one side of the equation, what must you do to the other side?

Correct Answer: Add the same value

Question 5:

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

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

Question 6:

What is the vertex of the quadratic equation y = (x - 3)² + 5?

Correct Answer: (3, 5)

Question 7:

In completing the square, the expression x² + 6x + ___ creates a perfect square trinomial.

Correct Answer: 9

Question 8:

If a quadratic equation in standard form is y = 2x² - 8x + 5, what value needs to be factored out of the x² and x terms before completing the square?

Correct Answer: 2

Question 9:

Which of the following quadratic equations is in vertex form?

Correct Answer: y = 2(x - 1)² + 3

Question 10:

When changing from standard form to vertex form by completing the square, the value of 'a' ______.

Correct Answer: Stays the same

Fill in the Blank Questions

Question 1:

The process of rewriting a quadratic equation to vertex form is called completing the ________.

Correct Answer: square

Question 2:

In vertex form, the ________ of the parabola is easily identifiable.

Correct Answer: vertex

Question 3:

Before completing the square when 'a' is not 1, you must ________ out the leading coefficient from the x² and x terms.

Correct Answer: factor

Question 4:

To maintain balance in the equation, whatever you add to one side, you must also add to the ________ side.

Correct Answer: other

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:

In completing the square for x² - 10x + c, the value of c that makes it a perfect square trinomial is ________.

Correct Answer: 25

Question 7:

The shortcut to factoring the perfect square trinomial after completing the square is to take half of the coefficient of the ________ term.

Correct Answer: x

Question 8:

When moving a constant from one side of the equation to the other, you perform the ________ operation.

Correct Answer: opposite

Question 9:

If the vertex form of a quadratic is y = (x + 4)² - 7, the x-coordinate of the vertex is ________.

Correct Answer: -4

Question 10:

The value 'a' in vertex form determines if the parabola opens upwards (a > 0) or ________ (a < 0).

Correct Answer: downwards

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.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. A2.A.2:Generate and evaluate equivalent algebraic expressions and equations using various strategies.

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