1. Lesson 3

1.4. Discover

Mathematics 20-1 Module 3

Module 3: Quadratic Functions

 

Discover

 

Try This 1

 

Complete the square using algebra tiles.

 

Step 1: Open Trinomial. Read the definition and examples and then scroll down to Demonstration Applet. In the applet the simplified form of the trinomial at the bottom represents the algebra tiles shown. Your task is to drag tiles into the workspace and form a perfect square.

 

 
This is a play button that opens Trinomial.



Step 2: Begin with the trinomial that is presented in Trinomial Demonstration Applet, which you accessed in Step 1. The trinomial is x2 + 2x − 8. Hold down the “r” key and click on an x-tile to rotate the tile. Move the x-tiles to form the partial grey square as shown.

 

 

This shows a partial grey square consisting of one x2-tile with a vertical x-tile on the right side and a horizontal x-tile along the base. There are 8 red negative-one tiles, which have not yet been used.

 

Step 3: To turn the grey x2 and x’s into a square, you will have to drag a +1 tile to the corner. When you are completing a square, every time you drag in a positive tile, you have to drag in a corresponding negative tile. This ensures that the value of the simplified trinomial remains the same. To keep your equation balanced, drag in a −1 tile now.

 

 

The partial grey square from Step 2 has been completed by adding a plus-one tile on the lower right. A red negative-one tile has been added to the eight red negative-one tiles.

 

Note: x2 + 2x + 1 is a perfect square since it has one factor that is squared, (x + 1)2.

 

 

This illustration shows a square of side (x + 1), and 9 negative-1 tiles. The lengths of the sides of the square are labelled (x + 1). (x + 1) squared is shown as a perfect square or x squared + 2x + 1.

 

Therefore, the original given equation, x2 + 2x − 8 may be written as (ax2 + bx + k) − 9. This is similar to the unsimplified form at the bottom of the applet with the zero-coefficient terms deleted.

 

 

This illustration shows a square of side (x + 1) and 9 negative-1 tiles. The lengths of the sides of the square are labelled (x + 1). (x + 1) squared − 9 is shown as x squared + 2x + 1 − 9.

 

Step 4: Complete the following tasks for each of the original trinomials shown in the chart and some trinomials that you make up:

  • Begin with the first two terms of the trinomial and complete the square by dragging in positive squares. Record the positive value added.
  • Bring in corresponding negative squares to balance the positive squares brought in. Record the total value left over.
  • Write the trinomial in unsimplified form.
  • Take screenshots or make sketches of the perfect square and the leftover value.
  1. Record your observations in a chart like the one shown here.

     
    Original Trinomial Positive Value Added Perfect-Square Trinomial

    Total Value

    Left Over

    		 Trinomial Written as the Unsimplified Form
    x2 + 2x − 8 1

    x2 + 2x + 1

     

    This illustration shows a square of side (x + 1).

    −9

     

    This illustration shows nine negative-one tiles.

    		 (x2 + −2x + 1) − 9
    x2 + 4x − 8        
    x2 + −8x + 5        
    x2 + 6x + 4        


  2. What relationship do you see between the coefficient of the second term in the original trinomial and the positive value added? hint hint

course folder Save your responses in your course folder.

 

Share 1

 

Based on your observations from Try This 1, discuss the following questions with a partner or group.

  1. What is the relationship between half of the coefficient of the second term in a trinomial and the number added to complete the square? Describe the patterns you found.

  2. Summarize your discussion by creating a general rule about the number added to complete a square and the value of b in y = ax2 + bx + c.

course folder Save your responses in your course folder.

Look at the relationships in all the trinomials to see the pattern. Does a pattern become clearer if you look at the relationship between half of the coefficient of the second term in the trinomial and what was added?
In the first original trinomial, x2 + 2x − 8, what is the relationship between the “2” in 2x and the “1” that was added.