Today we'll explore the algebraic expression (a+b)², which represents 'a plus b, all squared'. This fundamental expression can be understood in two complementary ways: through algebraic manipulation using the distributive property, and through geometric visualization using area models. We'll see how both approaches lead to the same expanded form: a² + 2ab + b².
Let's derive the expansion algebraically using the distributive property. We start with (a+b)² and rewrite it as (a+b)(a+b). Then we apply the FOIL method: First terms give us a², Outer terms give ab, Inner terms give another ab, and Last terms give b². Combining the like terms ab plus ab equals 2ab, we get our final result: a² + 2ab + b².
Now let's visualize this geometrically. We create a square with side length (a+b). By drawing internal lines, we divide this square into four distinct regions. The top-left region is a square with area a², the bottom-right is a square with area b², and the two remaining rectangles each have area ab. The total area of the large square equals the sum of these four regions.
Let's verify that our geometric model gives the same result as our algebraic expansion. We calculate the area of each colored region: the red square contributes a², the yellow square contributes b², and each of the two rectangles contributes ab. Adding these together: a² plus ab plus ab plus b² equals a² plus 2ab plus b². This perfectly matches our algebraic result!
Let's verify with a concrete example where a equals 3 and b equals 2. Algebraically, (3+2)² equals 5² which is 25. Using our expansion formula: 3² plus 2 times 3 times 2 plus 2² equals 9 plus 12 plus 4, which is 25. Geometrically, our square has side length 5, with areas 9, 6, 6, and 4, totaling 25. Both methods give the same result!