Difference of Squares Identity

A square plaza with side length x has a total area of x-squared. There is a smaller square section of greenery in the corner, with side length y and area y-squared. The remaining area of the plaza, which is paved, is the difference between the areas of the two squares.

An alternative way to calculate this L-shaped area is by using the difference of squares identity, which is x-squared minus y-squared equals the product of x plus y and x minus y.

To prove this, imagine making a horizontal cut and rotating the cut section so the pieces form a single rectangle. This transforms the L-shaped area into a single rectangle, making it much easier to measure.

The width of the new rectangle is x minus y, because a section of length y was removed from one side.

The length of the new rectangle is x plus y, because the same section was added to the bottom.

The area of this new rectangle is the product of its sides, which is exactly equal to the initial area of the L-shaped plaza, proving the difference of squares identity.

This concept also simplifies numerical calculations. For example, twenty-five squared minus five squared, using the factored form, gives six hundred.