Welcome to our exploration of formulas involving a squared plus b squared. While this expression doesn't have a single defining formula, it appears in several important algebraic identities. Today we'll examine the two main formulas that express a squared plus b squared in terms of other quantities.
Let's derive the first formula step by step. We start with the well-known expansion of a plus b squared, which equals a squared plus two a b plus b squared. To isolate a squared plus b squared, we simply subtract two a b from both sides, giving us our first formula: a squared plus b squared equals a plus b squared minus two a b.
Now let's derive the second formula. We start with the expansion of a minus b squared, which equals a squared minus two a b plus b squared. To isolate a squared plus b squared, we add two a b to both sides, giving us our second formula: a squared plus b squared equals a minus b squared plus two a b.
Let's verify our formulas with a numerical example. Using a equals 3 and b equals 4, we can calculate a squared plus b squared directly as 9 plus 16, which equals 25. Using our first formula, a plus b squared minus two a b gives us 49 minus 24, which also equals 25. Using our second formula, a minus b squared plus two a b gives us 1 plus 24, which again equals 25. All three methods produce the same result, confirming our formulas are correct.
To summarize what we've learned: There are two main formulas that express a squared plus b squared in terms of other quantities. The first formula states that a squared plus b squared equals a plus b squared minus two a b. The second formula states that a squared plus b squared equals a minus b squared plus two a b. Both formulas are derived from binomial expansions and are very useful in algebraic manipulations and problem solving.