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Algebra ▼

Quadratic Formula

x = [-b ± √(b² – 4ac)] / 2a

Solutions to the quadratic equation ax² + bx + c = 0

Binomial Theorem

(a + b)ⁿ = Σ [n!/(k!(n-k)!] aⁿ⁻ᵏbᵏ

Expansion of (a + b) raised to any positive integer power n

Arithmetic Series

Sₙ = n/2 (a₁ + aₙ) = n/2 [2a₁ + (n-1)d]

Sum of the first n terms of an arithmetic sequence

Geometry ▼

Pythagorean Theorem

a² + b² = c²

Relationship between sides of a right triangle

Area of Circle

A = πr²

Area of a circle with radius r

Volume of Sphere

V = (4/3)πr³

Volume of a sphere with radius r

Trigonometry ▼

Pythagorean Identity

sin²θ + cos²θ = 1

Fundamental relationship between sine and cosine

Law of Cosines

c² = a² + b² – 2ab cos(C)

Relates lengths of sides to cosine of included angle

Double Angle Formula

sin(2θ) = 2 sinθ cosθ

Sine of double angle in terms of single angle

Calculus ▼

Power Rule

d/dx(xⁿ) = nxⁿ⁻¹

Derivative of x raised to any real power n

Fundamental Theorem of Calculus

∫ₐᵇ f(x) dx = F(b) – F(a)

Relates differentiation and integration

Integration by Parts

∫ u dv = uv – ∫ v du

Technique for integrating products of functions

Probability ▼

Bayes’ Theorem

P(A|B) = P(B|A)P(A) / P(B)

Conditional probability relationship

Binomial Probability

P(k) = C(n,k) pᵏ (1-p)ⁿ⁻ᵏ

Probability of k successes in n independent trials

Statistics ▼

Normal Distribution

f(x) = (1/σ√2π) e⁻⁽ˣ⁻μ⁾²⁄²σ²

Probability density function for normal distribution

Standard Deviation

σ = √[Σ(xᵢ – μ)² / N]

Measure of dispersion in a dataset