Today we'll explore the difference of squares formula: a squared minus b squared. This is one of the most important algebraic identities. We start with a large square of side length a, which has area a squared. Then we remove a smaller square of side length b from the corner, with area b squared. The remaining area is a squared minus b squared.
To understand how this factors, we need to cut the L-shaped remaining area into rectangles. We make a horizontal cut from the top-left corner of the removed square straight across to the left edge of the large square. This creates two rectangles: a tall rectangle on the left with dimensions (a minus b) by a, and a smaller rectangle at the bottom with dimensions b by (a minus b).
Now comes the key insight. We can rearrange these two rectangles to form a single larger rectangle. When we place them together, the width of the new rectangle is (a minus b), and the height is (a plus b). Therefore, the total area is (a minus b) times (a plus b). Since we haven't changed the total area, this proves that a squared minus b squared equals (a minus b)(a plus b).
Let's verify this result algebraically. Starting with (a minus b)(a plus b), we use the distributive property. First, we distribute a to get a squared plus ab, then distribute negative b to get minus ba minus b squared. Since ab and ba are the same, they cancel out, leaving us with a squared minus b squared. This confirms our geometric insight: a squared minus b squared equals (a minus b)(a plus b).
The difference of squares formula has many practical applications in algebra. For example, x squared minus 9 factors as (x minus 3)(x plus 3), since 9 equals 3 squared. Similarly, 25 minus y squared becomes (5 minus y)(5 plus y). For more complex expressions like 4a squared minus 16b squared, we first factor out the common factor 4, then apply the difference of squares to get 4(a minus 2b)(a plus 2b). Remember to always look for this pattern: a squared minus b squared equals (a minus b)(a plus b).