## Proof Without Words: Completing the Square

The act of "completing the square" involves taking half the coefficient of *x* in the quadratic *x*^{2} + *ax* and adding its square. But many students do not understand why this process works. This interactive geometric proof shows…

**Why is**

*x*^{2}+*ax*= (*x*+*a*/2)^{2}– (*a*/2)^{2}?- Use the slider (red dot) 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
*x*^{2}+*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?