Square of a sum. Square of a difference
(a + b)² = (a + b)(a + b) = a² + 2ab + b² is the square of a sum: add the square of the first number, the double product of the first and second numbers, and the square of the second. (a - b)² = a² - 2ab + b² is the square of a difference, whose middle term is negative. These equalities are called short multiplication formulas. Geometrically, a square of side a + b consists of two squares with sides a and b and two rectangles of area ab. The formulas also help to square numbers quickly: 52² = (50 + 2)² = 2704. Used in reverse, they factor expressions such as x² + 10x + 25 = (x + 5)². The cube of a sum and of a difference: (a + b)³ = a³ + 3a²b + 3ab² + b³, (a - b)³ = a³ - 3a²b + 3ab² - b³.
“Building the square”: on squared paper a square of side a + b (for example 3 + 2 squares) is drawn and split into four parts. Pupils add the areas of the parts and “see” the formula a² + 2ab + b².