Perfect Square Trinomial: Square of a Sum

Imagine an engineer designing a square solar panel array with a side length of a. The total area covered by the panels is a squared.

As energy demand grows, the engineer decides to expand the array to generate more power. He extends the array by b units along two adjacent edges. The new array now has a side length of a plus b.

To find the area of the extended array, the engineer looks at the sections created by the expansion.

The original array remains a square with an area of a squared. The extension adds two identical rectangles along the edges, each with an area of ab.

Where these rectangles meet, a small square fills the gap. This square has a side length of b and an area of b squared.

Adding all these regions gives a squared plus 2ab plus b squared.

The entire solar array is also one large square with a side length of a plus b. Its area is a plus b, whole squared.

Since both expressions represent the area of the same array, they must be equal. This shows how a three-term trinomial factors into a perfect square.