Start with a square of side length and add two rectangles, each with side lengths and . The total area is . Place the rectangles along two adjacent sides of the square: the missing corner is a by square, of area . Filling it produces a square of side length .
Figure 1.
Two rectangles and a missing corner explain
. An original area assembly, drawn at the stipulated positive side length . The corner has area ; the remaining constant gives . The diagram uses positive side lengths, while the polynomial identity holds for every in the real numbers. Details.
The area picture explains why the half-coefficient is squared. It also separates two operations that are easy to confuse: adding changes , whereas adding and subtracting preserves the expression. When solving an equation instead, adding to one side requires adding to the other side too.

Ancestors (6)

  1. Completing the square
  2. Polynomial
  3. Algebra
  4. Area of mathematics
  5. Mathematics
  6. Home

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