An op-ed at the New York Times questioned the necessity of Algebra. At one point the writer wonders:
What of the claim that mathematics sharpens our minds and makes us more intellectually adept as individuals and a citizen body? It’s true that mathematics requires mental exertion. But there’s no evidence that being able to prove (x² + y²)² = (x² – y²)² + (2xy)² leads to more credible political opinions or social analysis.
And I thought, Maybe, but maybe the reverse is true — are people with more credible political opinions and social analysis — such as myself! — more likely able to prove the two sides are equivalent?
Forget about the second term on the right hand side of equation #1 for a second. When you carry out the binomial expansion of the sum of the squares (left), and difference of the squares (right), only the middle term is different. Do this and prove it to yourself.
The middle terms of the left and right expansion appear in #2. Clearly the term on the left is greater than the one on the right by 4 units. In other words, to make the two binomial expansions equivalent, we need to add 4 units of (x² • y²) to the right side.
Now back to that second term. Carry out the square and you see it equals 4 times (x² • y²) (#3).
Which is exactly what we just showed we need to add to the right side to make it equal the left (#4).
Done. Q.E.D.