Completing the Square: Unlocking Standard Form & Vertex Secrets

Algebra 2 Grades High School 5:47 Video
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Lesson Description

Master the technique of completing the square to transform quadratic equations into standard form, revealing the vertex and key properties. This lesson covers various difficulty levels, including factoring out leading coefficients.

Video Resource

Write a Quadratic Equation in Standard Form by Completing the Square (3 Examples)

Mario's Math Tutoring

Duration: 5:47
Watch on YouTube

Key Concepts

  • Standard form of a quadratic equation: y = a(x - h)^2 + k, where (h, k) is the vertex.
  • Completing the square technique.
  • Factoring out the leading coefficient when a ≠ 1.
  • Identifying the vertex from the standard form of a quadratic equation.

Learning Objectives

  • Students will be able to transform a quadratic equation from general form to standard form by completing the square.
  • Students will be able to identify the vertex of a quadratic equation once it's in standard form.
  • Students will be able to factor out a leading coefficient before completing the square.

Educator Instructions

  • Introduction (5 mins)
    Briefly review the general form of a quadratic equation (ax^2 + bx + c = 0) and introduce the concept of standard form (y = a(x - h)^2 + k). Explain why converting to standard form is useful (identifying the vertex).
  • Completing the Square - Example 1 (10 mins)
    Work through the first example from the video (easy example). Emphasize the steps: moving the constant term, taking half of the 'b' coefficient, squaring it, and adding it to both sides of the equation. Show how to factor the perfect square trinomial and rewrite the equation in standard form. Identify the vertex.
  • Completing the Square with a Leading Coefficient - Example 2 (15 mins)
    Work through the second example from the video (medium example). Highlight the additional step of factoring out the leading coefficient before completing the square. Explain how adding a constant inside the parentheses affects the other side of the equation (due to the factored coefficient). Again, identify the vertex.
  • Challenging Example with Fractions - Example 3 (15 mins)
    Work through the third example from the video (most challenging example). Focus on dealing with fractions and negative signs when factoring and completing the square. Reinforce the concept of multiplying by the reciprocal when factoring out a fraction. Identify the vertex.
  • Practice & Review (10 mins)
    Provide students with practice problems of varying difficulty levels to work on independently or in small groups. Review the key steps and address any remaining questions.

Interactive Exercises

  • Whiteboard Practice
    Have students come to the whiteboard to solve quadratic equations by completing the square, explaining each step to the class.
  • Group Problem Solving
    Divide students into small groups and assign each group a different quadratic equation to convert to standard form and find the vertex.

Discussion Questions

  • Why is it important to add the same value to both sides of the equation when completing the square?
  • How does the 'a' value in the standard form of a quadratic equation affect the shape of the parabola?
  • What are some real-world applications of quadratic equations and their vertex?

Skills Developed

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

Multiple Choice Questions

Question 1:

What is the standard 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' coefficient?

Correct Answer: Divide it by 2 and square it

Question 3:

What does the vertex (h, k) represent in the standard form of a quadratic equation?

Correct Answer: Maximum or minimum point

Question 4:

If you factor out a '3' from the expression 3x^2 + 12x, what's inside the parenthesis?

Correct Answer: x^2 + 4x

Question 5:

Which of the following is the correct vertex for the quadratic equation y = (x - 2)^2 + 3?

Correct Answer: (2, 3)

Question 6:

What is the first step in completing the square for the equation y = x^2 + 6x + 5?

Correct Answer: Subtract 5 from both sides

Question 7:

In the standard form y = a(x - h)^2 + k, what does the 'a' value indicate?

Correct Answer: Vertical stretch/compression and direction of opening

Question 8:

If you are completing the square for y = 2x^2 + 8x + 1, what do you need to factor out first?

Correct Answer: 2

Question 9:

After completing the square, if you have y = (x + 3)^2 - 4, what is the vertex of the parabola?

Correct Answer: (-3, -4)

Question 10:

What is the value that completes the square for x^2 - 10x + ____?

Correct Answer: 25

Fill in the Blank Questions

Question 1:

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

Correct Answer: vertex

Question 2:

To complete the square for x^2 + 6x, you need to add (6/2)^2, which equals __________.

Correct Answer: 9

Question 3:

If a quadratic equation in vertex form is y = (x - 3)^2 + 5, then the x-coordinate of the vertex is __________.

Correct Answer: 3

Question 4:

When factoring out a negative from an expression, remember to change the __________ of all terms inside the parentheses.

Correct Answer: sign

Question 5:

In completing the square, the number you add to one side of the equation must also be added to the __________ side to maintain equality.

Correct Answer: other

Question 6:

The process of rewriting a quadratic equation to easily identify its vertex is called completing the __________.

Correct Answer: square

Question 7:

Before completing the square, if the coefficient of x^2 is not 1, you must __________ it out.

Correct Answer: factor

Question 8:

If you divide the b value by 2 and square it you are finding the value to __________ the square.

Correct Answer: complete

Question 9:

Given the vertex form y = a(x - h)^2 + k, the y-coordinate of the vertex is represented by the variable __________.

Correct Answer: k

Question 10:

When moving a constant to the other side of the equation, you perform the __________ operation.

Correct Answer: opposite

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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