Algebra

1. \((a + b)^2 = a^2 + 2ab + b^2 = (a - b)^2 + 4ab\)
2. \((a - b)^2 = a^2 - 2ab + b^2 = (a + b)^2 - 4ab\)
3. \(a^2 - b^2 = (a + b)(a - b)\)
4. \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 = a^3 + b^3 + 3ab(a + b)\)
5. \((a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 = a^3 - b^3 - 3ab(a - b)\)
6. \(a^3 + b^3 = (a + b)(a^2 - ab + b^2) = (a + b)^3 - 3ab(a + b)\)
7. \((a + b)^2 + (a - b)^2 = 2(a^2 + b^2)\)
8. \((a + b)^2 - (a - b)^2 = 4ab\)
9. \((a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca\)
10. \(\frac{a}{b} = \frac{c}{d} \Rightarrow \frac{a + c}{b + d}\)
11. If \(\frac{a}{b} = \frac{c}{d}\), then \(\frac{a + b}{a - b} = \frac{c + d}{c - d}\)