Formula Library

LaTeX Formula Library

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Showing 48 of 48 formulas

Quadratic Formula

Solution to quadratic equations ax² + bx + c = 0

Algebra

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}

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Binomial Theorem

Expansion of (x+y)ⁿ

Algebra

(x + y)^{n} = k = 0 \sum n (k n) x^{n - k} y^{k}

(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k

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Difference of Squares

Factorization of difference of two squares

Algebra

a^{2} - b^{2} = (a + b) (a - b)

a^2 - b^2 = (a+b)(a-b)

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Completing the Square

Method to convert quadratic to vertex form

Algebra

a x^{2} + b x + c = a (x + \frac{b}{2 a})^{2} + c - \frac{b ^{2}}{4 a}

ax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 + c - \frac{b^2}{4a}

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Geometric Series

Sum of finite geometric series

Algebra

k = 0 \sum n a r^{k} = a \frac{1 - r ^{n + 1}}{1 - r}, r \neq = 1

\sum_{k=0}^{n} ar^k = a\frac{1-r^{n+1}}{1-r}, \quad r \neq 1

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Arithmetic Series

Sum of arithmetic sequence

Algebra

k = 1 \sum n (a + (k - 1) d) = \frac{n}{2} (2 a + (n - 1) d)

\sum_{k=1}^{n} (a + (k-1)d) = \frac{n}{2}(2a + (n-1)d)

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Derivative Definition

Definition of derivative as a limit

Calculus

f^{'} (x) = h \to 0 lim \frac{f ( x + h ) - f ( x )}{h}

f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

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Definite Integral

Fundamental theorem of calculus

Calculus

\int_{a}^{b} f (x) d x = F (b) - F (a)

\int_{a}^{b} f(x) \, dx = F(b) - F(a)

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Integration by Parts

Product rule for integration

Calculus

\int u d v = uv - \int v d u

\int u \, dv = uv - \int v \, du

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Chain Rule

Derivative of composite functions

Calculus

\frac{d}{d x} [f (g (x))] = f^{'} (g (x)) \cdot g^{'} (x)

\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)

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Product Rule

Derivative of product of functions

Calculus

\frac{d}{d x} [f (x) g (x)] = f^{'} (x) g (x) + f (x) g^{'} (x)

\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)

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Quotient Rule

Derivative of quotient of functions

Calculus

\frac{d}{d x} [\frac{f ( x )}{g ( x )}] = \frac{f ^{'} ( x ) g ( x ) - f ( x ) g ^{'} ( x )}{[ g ( x ) ] ^{2}}

\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}

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Power Rule

Derivative of power function

Calculus

\frac{d}{d x} [x^{n}] = n x^{n - 1}

\frac{d}{dx}[x^n] = nx^{n-1}

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Taylor Series

Taylor series expansion around point a

Calculus

f (x) = n = 0 \sum \infty \frac{f ^{(n)} ( a )}{n !} (x - a)^{n}

f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n

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L'Hôpital's Rule

Evaluate limits of indeterminate forms

Calculus

x \to c lim \frac{f ( x )}{g ( x )} = x \to c lim \frac{f ^{'} ( x )}{g ^{'} ( x )}

\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}

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Mean (Average)

Arithmetic mean of a dataset

Statistics

\overset{x}{ˉ} = \frac{1}{n} i = 1 \sum n x_{i}

\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i

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Variance

Measure of data spread

Statistics

σ^{2} = \frac{1}{n} i = 1 \sum n (x_{i} - μ)^{2}

\sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2

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Standard Deviation

Square root of variance

Statistics

σ = \frac{1}{n} i = 1 \sum n (x_{i} - μ)^{2}

\sigma = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2}

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Normal Distribution

Gaussian distribution probability density

Statistics

f (x) = \frac{1}{σ 2 π} e^{- \frac{1}{2} (\frac{x - μ}{σ})^{2}}

f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}

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Covariance

Measure of joint variability

Statistics

Cov (X, Y) = \frac{1}{n} i = 1 \sum n (x_{i} - \overset{x}{ˉ}) (y_{i} - \overset{y}{ˉ})

\text{Cov}(X,Y) = \frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})

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Correlation Coefficient

Pearson correlation coefficient

Statistics

r = \frac{Cov ( X , Y )}{σ _{X} σ _{Y}}

r = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y}

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Bayes' Theorem

Update probability based on new evidence

Statistics

P (A ∣ B) = \frac{P ( B ∣ A ) P ( A )}{P ( B )}

P(A|B) = \frac{P(B|A)P(A)}{P(B)}

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Binomial Probability

Probability of k successes in n trials

Statistics

P (X = k) = (k n) p^{k} (1 - p)^{n - k}

P(X = k) = \binom{n}{k}p^k(1-p)^{n-k}

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Matrix Multiplication

Definition of matrix product

Linear Algebra

(A B)_{ij} = k = 1 \sum n A_{ik} B_{kj}

(AB)_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj}

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2×2 Determinant

Determinant of 2×2 matrix

Linear Algebra

det (a c b d) = a d - b c

\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc

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Dot Product

Inner product of two vectors

Linear Algebra

a \cdot b = i = 1 \sum n a_{i} b_{i} = ∣ a ∣∣ b ∣ cos θ

\vec{a} \cdot \vec{b} = \sum_{i=1}^{n} a_i b_i = |\vec{a}||\vec{b}|\cos\theta

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Cross Product

Vector product in 3D space

Linear Algebra

a \times b = i a_{1} b_{1} j a_{2} b_{2} k a_{3} b_{3}

\vec{a} \times \vec{b} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}

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Eigenvalue Equation

Matrix eigenvalue and eigenvector

Linear Algebra

A v = λ v

A\vec{v} = \lambda\vec{v}

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Matrix Trace

Sum of diagonal elements

Linear Algebra

tr (A) = i = 1 \sum n a_{ii}

\text{tr}(A) = \sum_{i=1}^{n} a_{ii}

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Pythagorean Identity

Fundamental trigonometric identity

Trigonometry

sin^{2} θ + cos^{2} θ = 1

\sin^2\theta + \cos^2\theta = 1

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Law of Sines

Relationship between sides and angles in triangles

Trigonometry

\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C}

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

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Law of Cosines

Generalization of Pythagorean theorem

Trigonometry

c^{2} = a^{2} + b^{2} - 2 ab cos C

c^2 = a^2 + b^2 - 2ab\cos C

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Double Angle (Sine)

Sine of double angle

Trigonometry

sin (2 θ) = 2 sin θ cos θ

\sin(2\theta) = 2\sin\theta\cos\theta

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Double Angle (Cosine)

Cosine of double angle

Trigonometry

cos (2 θ) = cos^{2} θ - sin^{2} θ

\cos(2\theta) = \cos^2\theta - \sin^2\theta

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Sum of Angles

Sine of sum of two angles

Trigonometry

sin (α + β) = sin α cos β + cos α sin β

\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta

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Euler's Formula

Complex exponential and trigonometric functions

Trigonometry

e^{i θ} = cos θ + i sin θ

e^{i\theta} = \cos\theta + i\sin\theta

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Newton's Second Law

Force equals mass times acceleration

Physics

F = ma

F = ma

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Mass-Energy Equivalence

Einstein's famous equation

Physics

E = m c^{2}

E = mc^2

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Kinetic Energy

Energy of motion

Physics

K E = \frac{1}{2} m v^{2}

KE = \frac{1}{2}mv^2

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Schrödinger Equation

Fundamental equation of quantum mechanics

Physics

i ℏ \frac{\partial}{\partial t} Ψ = \hat{H} Ψ

i\hbar\frac{\partial}{\partial t}\Psi = \hat{H}\Psi

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Gravitational Potential Energy

Energy due to position in gravitational field

Physics

PE = m g h

PE = mgh

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Momentum

Linear momentum of an object

Physics

p = m v

p = mv

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Work-Energy Theorem

Relationship between work and kinetic energy

Physics

W = Δ K E = \frac{1}{2} m (v_{f}^{2} - v_{i}^{2})

W = \Delta KE = \frac{1}{2}m(v_f^2 - v_i^2)

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Ohm's Law

Relationship between voltage, current, and resistance

Physics

V = I R

V = IR

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Coulomb's Law

Electrostatic force between charges

Physics

F = k \frac{q _{1} q _{2}}{r ^{2}}

F = k\frac{q_1 q_2}{r^2}

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Universal Gravitation

Newton's law of universal gravitation

Physics

F = G \frac{m _{1} m _{2}}{r ^{2}}

F = G\frac{m_1 m_2}{r^2}

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Wave Equation

Relationship between wave speed, frequency, and wavelength

Physics

v = f λ

v = f\lambda

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Lorentz Force

Force on charged particle in electromagnetic field

Physics

F = q (E + v \times B)

\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})

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