Need help with mathematics?
Quadratic Formula
x = (-b ± √(b² - 4ac)) / 2a
Solves quadratic equations in the form ax² + bx + c = 0
Pythagorean Theorem
a² + b² = c²
Relates the lengths of the sides in a right triangle
Exponential Growth
A = P(1 + r)ᵗ
Models growth where rate of change is proportional to current value
Logarithmic Formula
logₐ(x) = y ⟺ aʸ = x
Expresses logarithmic relationships and their exponential equivalents
Distance Formula
d = √[(x₂-x₁)² + (y₂-y₁)²]
Distance between two points in a 2D plane
Slope Formula
m = (y₂-y₁)/(x₂-x₁)
Rate of change between two points
Compound Interest
A = P(1 + r/n)^(nt)
Growth of investment with interest compounded n times per year
Binomial Theorem
(x + y)ⁿ = Σᵏ₌₀ⁿ (ⁿCₖ) xⁿ⁻ᵏyᵏ
Expansion of binomial expressions with integer exponents
Arithmetic Sequence
aₙ = a₁ + (n - 1)d
Finding the nth term in an arithmetic progression
Combinations Formula
ⁿCᵣ = n! / (r! * (n-r)!)
Number of ways to choose r items from n items
Gaussian Distribution
f(x) = (1 / (σ * √(2π))) * e^(-(x-μ)^2 / (2σ^2))
Probability density function of the normal distribution
Golden Ratio
φ = (1 + √5) / 2 ≈ 1.618
Ratio found in many natural and man-made structures
Bayes' Theorem
P(A|B) = (P(B|A) * P(A)) / P(B)
Calculating conditional probabilities
Stirling's Formula
n! ≈ √(2πn) * (n/e)^n
Approximation of the factorial function
Cauchy-Schwarz Inequality
(Σ a_i b_i)² ≤ (Σ a_i²) * (Σ b_i²)
Relationship between dot product and vector magnitudes
Fourier Series
f(x) = a_0/2 + Σ (a_n cos(nx) + b_n sin(nx))
Represents a periodic function as the sum of sinusoidal functions
Gamma Function
Γ(x) = ∫₀^∞ t^(x-1) e^(-t) dt
Extension of the factorial function to non-integer values
Euler's Totient Function
φ(n) = n ∏ (1 - 1/p)
Number of positive integers less than or equal to n that are relatively prime to n