Today we will prove one of the most fundamental algebraic identities: the perfect square formula. We'll show that when we square the sum of two terms a plus b, we get a squared plus 2ab plus b squared. Let's start by visualizing this geometrically using a square diagram.
Now let's prove this formula algebraically. We start with (a+b)² which by definition equals (a+b) times (a+b). Using the distributive property, we can expand this as a times (a+b) plus b times (a+b). Continuing to distribute, we get a times a plus a times b plus b times a plus b times b. This simplifies to a² plus ab plus ba plus b². Since multiplication is commutative, ab equals ba, so we can combine these terms to get a² plus 2ab plus b².
Let's use the FOIL method to systematically expand (a+b)². FOIL stands for First, Outer, Inner, Last. First, we multiply the first terms: a times a equals a². Then the outer terms: a times b equals ab. Next the inner terms: b times a equals ba. Finally the last terms: b times b equals b². Combining all terms gives us a² plus ab plus ba plus b², which simplifies to a² plus 2ab plus b².
Let's verify our formula with a numerical example. If we let a equal 3 and b equal 2, then the left side becomes (3 + 2)² which equals 5² or 25. On the right side, we have 3² plus 2 times 3 times 2 plus 2², which equals 9 plus 12 plus 4, giving us 25. Since both sides equal 25, our formula is verified for this example.
We have successfully proven that (a+b)² equals a² plus 2ab plus b². We demonstrated this using three different methods: geometric visualization with area squares, algebraic expansion using the distributive property, and the FOIL method for systematic multiplication. This fundamental formula is essential in algebra for polynomial expansion, in calculus for derivatives, in physics and engineering calculations, and in geometry for area calculations. Understanding this identity opens the door to more advanced mathematical concepts.