Algebra identities

Square of a sum

(a + b)² = a² + 2ab + b²

Square of a difference

(a − b)² = a² − 2ab + b²

Difference of squares

a² − b² = (a + b)(a − b)

Cube of a sum

(a + b)³ = a³ + b³ + 3ab(a + b)

Cube of a difference

(a − b)³ = a³ − b³ − 3ab(a − b)

Sum of cubes

a³ + b³ = (a + b)(a² − ab + b²)

Three-variable

a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca)

If a + b + c = 0, a³ + b³ + c³ = 3abc

Quadratic roots

x = [−b ± √(b² − 4ac)] / 2a

D = b² − 4ac decides the roots

Sum & product of roots

α + β = −b/a, αβ = c/a

AP nth term & sum

aₙ = a + (n − 1)d, Sₙ = n/2[2a + (n − 1)d]

GP nth term & sum

aₙ = arⁿ⁻¹, Sₙ = a(rⁿ − 1)/(r − 1)

Laws of exponents

aᵐ·aⁿ = aᵐ⁺ⁿ; aᵐ/aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ

a⁰ = 1

Laws of logarithms

log(mn) = log m + log n; log(m/n) = log m − log n; log mⁿ = n log m

Change of base

logₐ b = log b / log a