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Proof Without Words: Completing the Square

Math Content:

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?

How to Use

  • You can change the side length of the square by adjusting the value inside the “x=” window.
  • Use the red points on the slider to adjust the width of the blue rectangle.
  • You will not be able to drag any objects within this workspace.


The buttons are meant to be used sequentially, and will appear in the order they are meant to be pressed.

Do not press buttons before all movements are complete.

The Reset button can be used at any time to start the demonstration over.

  • Explain in words how this proof works.
  • Construct a formal proof on why x2+ax=(x+a/2)2-(a/2)2.
  • How is completing the square similar to the quadratic equation?