Welcome to our lesson on the difference of squares! This is a fundamental algebraic pattern that appears frequently in mathematics. The difference of squares occurs when we subtract one perfect square from another perfect square, and it has a beautiful factoring formula that we'll explore together.
Now let's understand why this formula works. When we multiply (a minus b) times (a plus b), we use the distributive property. First, we get a squared plus a b minus b a minus b squared. Notice that the middle terms, positive a b and negative a b, are opposites and cancel each other out, leaving us with a squared minus b squared.
Let's work through some examples. First, x squared minus 9. We recognize this as x squared minus 3 squared, which factors to (x minus 3)(x plus 3). Second example: 25 minus y squared equals 5 squared minus y squared, which factors to (5 minus y)(5 plus y). Third example: 4x squared minus 16 equals (2x) squared minus 4 squared, factoring to (2x minus 4)(2x plus 4).
Here's a beautiful geometric interpretation. We start with a large square of side length 'a' and remove a smaller square of side length 'b' from one corner. The remaining area represents a squared minus b squared. Now, if we rearrange this remaining area, we can form two rectangles: one with dimensions (a minus b) by b, and another with dimensions a by (a minus b). This gives us the factored form: (a minus b) times (a plus b).
To summarize, the difference of squares is a fundamental algebraic pattern with the formula a squared minus b squared equals (a minus b)(a plus b). This powerful tool has many applications: factoring polynomials, solving quadratic equations, simplifying rational expressions, and even performing mental math calculations. Remember to always look for this pattern when you see two perfect squares being subtracted. With practice, recognizing and applying the difference of squares will become second nature in your algebraic toolkit.