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.
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!