Perfect Square Trinomial: Square of a Difference

Have you ever wondered how this specific trinomial factors into a square of a difference? Let’s look at an example to visualize this.

Imagine a square sheet of glass with side length a. Its total area is a squared. A technician wants to make a smaller square window. To fit the glass, he trims a strip of width b from two adjacent sides of the glass.

Each trimmed strip is a rectangle with an area of ab. Since two strips are removed, the original area decreases by 2ab.

The corner where the two strips overlap forms a square with an area of b squared.

While removing the strips, this square is counted twice in the subtraction. To correct this, the area b squared must be added back once. The remaining area is a squared minus 2ab plus b squared.

After the trimming, the glass is still a square with a side length of a minus b. So, the area of this smaller square is a minus b, whole squared.

Since both expressions represent the area of the same square, they must be equal. This shows how a trinomial factors into the square of a difference.