The difference of squares formula
The product of the difference of two numbers and their sum equals the difference of their squares: (a - b)(a + b) = a² - b². This can be checked by multiplying: a² + ab - ab - b², where the two middle terms cancel. The formula is also used from right to left: a² - b² = (a - b)(a + b), the formula for factoring a difference of squares. For example 25x² - 4y² = (5x)² - (2y)² = (5x - 2y)(5x + 2y). The role of a or b can be played by a bracket: (x + 1)² - 9 = (x + 1 - 3)(x + 1 + 3) = (x - 2)(x + 4). The formula helps in fast multiplication: 48 · 52 = (50 - 2)(50 + 2) = 2500 - 4 = 2496. Geometrically, if a small square is cut out of a large one, the remaining shape can be rearranged into a rectangle.
“Cutting a square”: an a × a square is drawn on squared paper and a b × b square is marked at a corner (use scissors only with an adult's help, or just draw the cut). The remaining shape is rearranged into an (a - b)(a + b) rectangle and the areas are compared.