In this section, you will use concrete models, or algebra tiles, to represent variables and constants. The algebra tiles will make it easier for you to understand the procedures used in completing the square.

To begin with, let’s define the algebra tiles that you will use.

What are the dimensions of the x² tile, and how do those dimensions compare to the tile used to represent x?

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The x² tile has a length of x units and a width of x units, which is the same as the length of the x tile.

The model below can be used to represent the polynomial x² + 8x

• What are the dimensions of the completed square in terms of x and a constant term?
  

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  Each dimension is x + 4 units.
• What expression containing a binomial that is squared would represent the area of the square if it was complete?
  

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  (x + 4)²
• Why do you think this method is called "completing the square?"
  

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  An additional number of units are required to geometrically complete the square in order to write the expression as a squared binomial.

Let's generalize what you saw with this one example.

Watch the following video:

Pause and Reflect

If the vertex form of the polynomial is (x - h)² + k and the general form of the polynomial is x² + bx + c, how are the variables h and k related to the variables b and c?

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If b is an odd number, how would using the method of completing the square be different from when b is an even number?

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If b is an even number, then when you divide b in half, you get a whole number. If b is an odd number, then you will have a decimal or mixed number. The procedures used in completing the square would be the same.

 

Practice

1. What number would you need to add to the expression x² – 8x in order to complete the square?

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   You will need to add the square of 1 over 2 1 2 of the coefficient of the x-term.

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   16
2. Use the method of completing the square to express x² - 17x in vertex form.

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   You will need to add the square of 1 over 2 1 2 of the coefficient of the x-term and then factor the resulting expression. Be sure to compensate for the added term by subtracting the same value.

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3. Use the method of completing the square to express x² + 20x = 15 in vertex form.

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   You will need to add the square of 1 over 2 1 2 of the coefficient of the x-term and then factor the resulting expression. Be sure to compensate for the added term by adding the same term to both sides of the equation.

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