Algebra: Key Formulas and Techniques

Important Identities

• (a + b)² = a² + 2ab + b²
• (a - b)² = a² - 2ab + b²
• a² - b² = (a + b)(a - b)
• (a + b)³ = a³ + 3a²b + 3ab² + b³
• a³ + b³ = (a + b)(a² - ab + b²)
• a³ - b³ = (a - b)(a² + ab + b²)

Special Patterns

• If x + 1/x = a, then x² + 1/x² = a² - 2
• If x + y = a and xy = b, then x² + y² = a² - 2b
• (x + y + z)² = x² + y² + z² + 2(xy + yz + zx)

Quadratic Equations

• For ax² + bx + c = 0:

- Sum of roots = -b/a

- Product of roots = c/a

- Discriminant = b² - 4ac

Logarithms

• log(ab) = log(a) + log(b)
• log(a/b) = log(a) - log(b)
• log(aⁿ) = n × log(a)
• logₐ(b) = 1/logᵦ(a)

CAT Tip

Most algebra questions test pattern recognition, not complex calculations.