Difference of Squares of Sum and Difference
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Theorem
- $\forall a, b \in \R: \paren {a + b}^2 - \paren {a - b}^2 = 4 a b$
Algebraic Proof
| \(\ds \) | \(\) | \(\ds \left({a + b}\right)^2 - \left({a - b}\right)^2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \left({a^2 + 2 a b + b^2}\right) - \left({a^2 - 2 a b + b^2}\right)\) | Square of Sum and Square of Difference | |||||||||||
| \(\ds \) | \(=\) | \(\ds a^2 + 2 a b + b^2 - a^2 + 2 a b - b^2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 2 a b + 2 a b\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 4 a b\) |
$\blacksquare$
