# Geometric Proof of the Difference of Squares: a² - b²

The **difference of two squares** is subtracting a square number from another squared number. And these numbers don’t have be perfect squares. Thankfully, difference of squares can be factored easily.

** a² - b²** is ubiquitous in mathematics and it is also supercalifragilisticexpialidocious for algebra.

If we speak algebraically:

*(a + b) (a - b) = a² + ba - ab -b²*

*= a² - b²*

But there is a different and beautiful way to represent ** a² - b²**. We can create multiple representations of this single concept. This is the beauty of mathematics. For instance geometric objects are so powerful to visualize algebraic formulas and equations.

***There is a beautiful book about geometrical proves of algebra that is called:The Elements of Euclid.You will love it if you have!

Let’s think about it geometrically a little bit.

This blue shape below has an area of ** a² - b²**. And we can reveal an algebraic identity by rearranging.

To do this, first we make a cut and split the shape into two different rectangles; the blue one and the yellow one. The height of the blue rectangle now is** (a - b)**, and the height of the yellow rectangle is obviously

**.**

*b*Now, if we flip the yellow rectangle and put next to the blue rectangle, we finish our rearranging. Since the area of a rectangle is height times width, the area of the combined rectangle is;

(a + b) ( a - b).

This rectangle has the same area as the original shape! Which means;

Sometimes representing an algebra problem geometrically can have interesting results!