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Proof Without Words: Completing the Square

Grade:
9-12
Standards:
Geometry
Math Content:
Geometry

The act of "completing the square" involves taking half the coefficient of x in the quadratic x2 + ax and adding its square. But many students do not understand why this process works. This interactive geometric proof shows…

Why is x2 + ax = (x + a/2)2 – (a/2)2?

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Use the slider to adjust the width of the blue rectangle.

Click Divide Rectangle in Half to bisect the blue rectangle. Then click Move Pieces and Align to place the pieces in a better arrangement around the yellow square.

Click Complete the Square to add the necessary piece to make the arrangement into a complete square.

The side length of the yellow square is x, and the width of the blue rectangle is a. (You can adjust the width of the rectangle using the slider at the top of the screen.)

  • What is the area of the yellow square?
  • What is the area of the blue rectangle?
  • How do these two shapes represent the expression x2 + ax?

The process of "completing the square" involves taking half the coefficient of x. Click the Divide Rectangle in Half button.

  • What are the dimensions of each of the smaller rectangles that are formed?
  • What is the area of each of these smaller rectangles?

Click the Move Pieces and the Align buttons to move the rectangles to a better arrangement. This new arrangement is almost, but not quite, a complete square.

  • If the square were completed, what would its side lengths be?
  • Click the Complete the Square button. In terms of a, what is the area of the orange square that has been added?

After the square has been completed, adjust the slider for the Rectangle Width (a).

  • As the width of the rectangle changes, how does the size of the orange square change?
  • In terms of a, what is the area of the orange square?