Quadratic Formula

Class 10algebra

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

Roots of a quadratic equation ax^2 + bx + c = 0

#roots#equations#quadratic

Difference of Squares

Class 10algebra

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

Factoring difference of two squares

#factoring#polynomials

Sum of Cubes

Class 10algebra

a^3 + b^3 = (a + b)(a^2 - ab + b^2)

Factoring sum of two cubes

#factoring#cubes

Difference of Cubes

Class 10algebra

a^3 - b^3 = (a - b)(a^2 + ab + b^2)

Factoring difference of two cubes

#factoring#cubes

Binomial Theorem

Class 12algebra

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

Expansion of a binomial power

#expansion#series#binomial

Arithmetic Progression (nth Term)

Class 10algebra

a_n = a_1 + (n-1)d

nth term of an AP with common difference d

#sequences#ap

Arithmetic Progression (Sum)

Class 10algebra

S_n = \frac{n}{2}(2a_1 + (n-1)d) = \frac{n}{2}(a_1 + a_n)

Sum of first n terms of an AP

#series#ap

Geometric Progression (nth Term)

Class 12algebra

a_n = a_1 r^{n-1}

nth term of a GP with common ratio r

#sequences#gp

Geometric Progression (Sum)

Class 12algebra

S_n = \frac{a_1(1-r^n)}{1-r} \quad (r \neq 1)

Sum of first n terms of a GP

#series#gp

Infinite Geometric Series

Class 12algebra

S_\infty = \frac{a_1}{1-r} \quad (|r| < 1)

Sum of infinite GP when ratio is strictly between -1 and 1

#series#infinite

Vieta's Formulas (Quadratic)

Class 10algebra

x_1 + x_2 = -\frac{b}{a}, \quad x_1 x_2 = \frac{c}{a}

Sum and product of roots of ax^2 + bx + c = 0

#roots#vieta#polynomials

Completing the Square

Class 10algebra

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

Rewriting quadratic into vertex form

#quadratic#factoring

Pythagorean Identity

Class 10trigonometry

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

Fundamental trigonometric identity relating sine and cosine

#identities#pythagorean

Secant & Tangent Identity

Class 10trigonometry

1 + \tan^2\theta = \sec^2\theta

Pythagorean identity for tangent and secant

#identities#pythagorean

Cosecant & Cotangent Identity

Class 10trigonometry

1 + \cot^2\theta = \csc^2\theta

Pythagorean identity for cotangent and cosecant

#identities#pythagorean

Sine Addition & Subtraction

Class 12trigonometry

\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B

Sine of sum or difference of two angles

#compound#addition

Cosine Addition & Subtraction

Class 12trigonometry

\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B

Cosine of sum or difference of two angles

#compound#addition

Tangent Addition & Subtraction

Class 12trigonometry

\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}

Tangent of sum or difference of two angles

#compound#addition

Double Angle (Sine)

Class 12trigonometry

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

Sine of double angle

#double-angle

Double Angle (Cosine)

Class 12trigonometry

\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta

Cosine of double angle formulas

#double-angle

Half Angle Formulas

Class 12trigonometry

\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1-\cos\theta}{2}}, \quad \cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1+\cos\theta}{2}}

Sine and cosine of half angle

#half-angle

Law of Sines

Class 10trigonometry

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R

Relates sides of a triangle to the sines of its angles

Interactive Triangle Solver (SAS):

Side a

Side b

Angle C (°)

Computed Side c:c = 5.568

#triangles#sines

Law of Cosines

Class 10trigonometry

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

Relates sides of a triangle to cosine of an angle

Interactive Triangle Solver (SAS):

Side a

Side b

Angle C (°)

Computed Side c:c = 5.568

#triangles#cosines

Power Rule (Derivative)

Calculuscalculus

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

Derivative of a power function

#derivative#power

Product Rule

Calculuscalculus

\frac{d}{dx}[u \cdot v] = u'v + uv'

Derivative of product of two functions

#derivative#product

Quotient Rule

Calculuscalculus

\frac{d}{dx}\left[\frac{u}{v}\right] = \frac{u'v - uv'}{v^2}

Derivative of quotient of two functions

#derivative#quotient

Chain Rule

Calculuscalculus

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

Derivative of composite functions

#derivative#chain

Derivative of Exponential

Calculuscalculus

\frac{d}{dx}[e^x] = e^x, \quad \frac{d}{dx}[a^x] = a^x \ln a

Derivative of natural and base-a exponential functions

#derivative#exponential

Derivative of Logarithm

Calculuscalculus

\frac{d}{dx}[\ln x] = \frac{1}{x}, \quad \frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}

Derivative of natural and base-a logarithms

#derivative#logarithm

Standard Trig Derivatives

Calculuscalculus

\frac{d}{dx}[\sin x] = \cos x, \quad \frac{d}{dx}[\cos x] = -\sin x, \quad \frac{d}{dx}[\tan x] = \sec^2 x

Derivatives of basic trigonometric functions

#derivative#trigonometry

Power Rule (Integral)

Calculuscalculus

\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)

Indefinite integral of power function

#integral#power

Reciprocal Integral

Calculuscalculus

\int \frac{1}{x} \, dx = \ln|x| + C

Integral of 1/x yielding natural logarithm

#integral#logarithm

Integration by Parts

Calculuscalculus

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

Standard integration technique for products

#integral#parts

Fundamental Theorem of Calculus

Calculuscalculus

\int_a^b f(x) \, dx = F(b) - F(a)

Evaluating definite integral from antiderivative F(x)

#integral#theorem

L'Hôpital's Rule

Calculuscalculus

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

Evaluating indeterminate forms (0/0 or ∞/∞)

#limit#lhopital

Product Rule for Logs

logarithms

\log_b(xy) = \log_b(x) + \log_b(y)

Logarithm of a product equals sum of logarithms

#product#logarithm

Quotient Rule for Logs

logarithms

\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)

Logarithm of a quotient equals difference of logarithms

#quotient#logarithm

Power Rule for Logs

logarithms

\log_b(x^k) = k \log_b(x)

Exponent comes out as a multiplier

#power#exponent#logarithm

Change of Base Formula

logarithms

\log_b(x) = \frac{\log_k(x)}{\log_k(b)} = \frac{\ln x}{\ln b} = \frac{\log_{10} x}{\log_{10} b}

Converting logarithm to a different base k

#base#conversion

Identity & Zero Properties

logarithms

\log_b(b) = 1, \quad \log_b(1) = 0

Basic identity properties of logarithms

#identity#logarithm

Inverse Properties

logarithms

b^{\log_b x} = x, \quad \log_b(b^x) = x

Logarithms and exponentials are inverse functions

#inverse#exponential

Natural Log & Euler Constant

logarithms

\ln(x) = \log_e(x), \quad e = \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^n \approx 2.71828

Definition of natural logarithm and base e

#natural#euler

Exponential Laws

logarithms

a^m \cdot a^n = a^{m+n}, \quad \frac{a^m}{a^n} = a^{m-n}, \quad (a^m)^n = a^{mn}

Fundamental laws of exponent arithmetic

#exponential#powers

Permutations Formula

probability

P(n, r) = {}^n P_r = \frac{n!}{(n-r)!}

Number of ordered arrangements of r elements from n

#combinatorics#permutations

Combinations Formula

probability

C(n, r) = {}^n C_r = \binom{n}{r} = \frac{n!}{r!(n-r)!}

Number of unordered selections of r elements from n

#combinatorics#combinations

Addition Rule of Probability

probability

P(A \cup B) = P(A) + P(B) - P(A \cap B)

Probability of union of two events

#probability#union

Multiplication Rule

probability

P(A \cap B) = P(A) \cdot P(B|A)

Probability of simultaneous occurrence of two events

#probability#intersection

Conditional Probability

probability

P(A|B) = \frac{P(A \cap B)}{P(B)} \quad (P(B) > 0)

Probability of event A occurring given event B has occurred

#probability#conditional

Bayes' Theorem

probability

P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} = \frac{P(B|A) P(A)}{\sum P(B|A_i) P(A_i)}

Calculates posterior probability based on prior knowledge

#probability#bayes

Binomial Distribution PMF

probability

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

Probability of k successes in n independent Bernoulli trials

#distribution#binomial

Poisson Distribution PMF

probability

P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}

Probability of k events in a fixed interval with average rate λ

#distribution#poisson

Expected Value (Discrete)

probability

E[X] = \sum_{i} x_i \cdot P(X = x_i)

Weighted average of all possible values of discrete random variable

#statistics#mean#expectation

Variance & Standard Deviation

probability

\text{Var}(X) = E[X^2] - (E[X])^2, \quad \sigma = \sqrt{\text{Var}(X)}

Measure of dispersion of a random variable

#statistics#variance

Standard Normal Distribution (PDF)

probability

f(z) = \frac{1}{\sqrt{2\pi}} e^{-z^2/2}, \quad z = \frac{x - \mu}{\sigma}

Probability density function of standard normal variable

#distribution#normal#z-score

Distance Formula (2D)

Class 10geometry

d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}

Distance between two points (x1, y1) and (x2, y2)

Interactive Solver:

Point 1 (x₁, y₁)

Point 2 (x₂, y₂)

Distance:5

Midpoint:(1.5, 2)

Slope (m):1.333

#coordinate#distance

Midpoint Formula

Class 10geometry

M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)

Midpoint of a line segment

#coordinate#midpoint

Slope of a Line

Class 10geometry

m = \frac{y_2-y_1}{x_2-x_1} = \tan\theta

Slope of line passing through two points

#coordinate#line#slope

Slope-Intercept Form

Class 10geometry

y = mx + c

Equation of a straight line with slope m and y-intercept c

#coordinate#line

Point-Slope Form

Class 12geometry

y - y_1 = m(x - x_1)

Equation of line passing through (x1, y1) with slope m

#coordinate#line

Standard Circle Equation

Class 12geometry

(x-h)^2 + (y-k)^2 = r^2

Circle with center (h, k) and radius r

#conic#circle

General Parabola Equation

Class 12geometry

y = a(x-h)^2 + k \quad \text{or} \quad (y-k)^2 = 4a(x-h)

Parabola in vertex form

#conic#parabola

Standard Ellipse Equation

Class 12geometry

\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1

Ellipse centered at (h, k) with semi-major/minor axes a, b

#conic#ellipse

Standard Hyperbola Equation

Class 12geometry

\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1

Horizontal hyperbola centered at (h, k)

#conic#hyperbola

Section Formula (Internal)

Class 10geometry

P = \left(\frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n}\right)

Coordinates of point dividing line segment in ratio m:n

Interactive Solver:

Point A (x₁, y₁)

Point B (x₂, y₂)

Ratio m : n

Point P:(2, 4)

#coordinate#section

Perpendicular Distance from Point to Line

Class 12geometry

d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}

Distance from point (x1, y1) to line Ax + By + C = 0

#coordinate#distance#line

Vector Dot Product

vectors

\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta = a_x b_x + a_y b_y + a_z b_z

Scalar product of two vectors

#vectors#dot-product

Vector Cross Product

vectors

\vec{a} \times \vec{b} = |\vec{a}||\vec{b}|\sin\theta \, \hat{n} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix}

Vector product yielding orthogonal vector

#vectors#cross-product

Vector Magnitude

vectors

|\vec{a}| = \sqrt{a_x^2 + a_y^2 + a_z^2}

Euclidean norm or length of a vector

#vectors#magnitude

Unit Vector

vectors

\hat{a} = \frac{\vec{a}}{|\vec{a}|}

Vector of length 1 pointing in same direction

#vectors#unit-vector

Scalar Projection of A onto B

vectors

\text{proj}_{\vec{b}}(\vec{a}) = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}

Magnitude of vector a in direction of vector b

#vectors#projection

Matrix Determinant (2x2)

vectors

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

Determinant of a 2 by 2 matrix

#matrices#determinant

Matrix Inverse (2x2)

vectors

\begin{bmatrix} a & b \\ c & d \end{bmatrix}^{-1} = \frac{1}{ad-bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}

Inverse of an invertible 2 by 2 matrix

#matrices#inverse

Characteristic Equation (Eigenvalues)

vectors

\det(A - \lambda I) = 0

Equation for finding eigenvalues λ of matrix A

#matrices#eigenvalues

Euler's Formula

complex

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

Links complex exponential with trig functions

#euler#polar

De Moivre's Theorem

complex

(r(\cos\theta + i\sin\theta))^n = r^n(\cos(n\theta) + i\sin(n\theta))

Power of a complex number in polar form

#power#polar

Modulus of Complex Number

complex

|z| = \sqrt{a^2 + b^2}

Absolute value of z = a + bi

#magnitude#modulus

Complex Conjugate

complex

\bar{z} = a - bi

Conjugate of z = a + bi

#conjugate

Polar Form

complex

z = r(\cos\theta + i\sin\theta)

Polar representation of a complex number

#polar#argument

Argument of Complex Number

complex

\arg(z) = \tan^{-1}\left(\frac{b}{a}\right)

Angle of z = a + bi in the complex plane

#argument#angle

Complex Multiplication

complex

(a+bi)(c+di) = (ac-bd) + (ad+bc)i

Product of two complex numbers

#multiplication#arithmetic

Complex Division

complex

\frac{a+bi}{c+di} = \frac{(ac+bd)+(bc-ad)i}{c^2+d^2}

Quotient of two complex numbers

#division#arithmetic

Roots of Unity

complex

z_k = e^{i \cdot 2\pi k/n}, \quad k=0,1,\ldots,n-1

The n-th roots of unity

#roots#unity

Exponential Form

complex

z = re^{i\theta}

Exponential representation using Euler formula

#exponential#polar

Circle Area

mensuration

A = \pi r^2

Area of a circle with radius r

#2d#area#circle

Circle Circumference

mensuration

C = 2\pi r

Perimeter of a circle

#2d#perimeter#circle

Triangle Area

mensuration

A = \frac{1}{2}bh

Area of a triangle with base b and height h

#2d#area#triangle

Heron's Formula

mensuration

A = \sqrt{s(s-a)(s-b)(s-c)}

Triangle area from side lengths, s = (a+b+c)/2

#2d#area#triangle

Rectangle Area

mensuration

A = l \times w

Area of a rectangle

#2d#area#rectangle

Trapezium Area

mensuration

A = \frac{1}{2}(a+b)h

Area of a trapezium with parallel sides a, b

#2d#area#trapezium

Sphere Volume

mensuration

V = \frac{4}{3}\pi r^3

Volume of a sphere

#3d#volume#sphere

Sphere Surface Area

mensuration

S = 4\pi r^2

Surface area of a sphere

#3d#surface#sphere

Cylinder Volume

mensuration

V = \pi r^2 h

Volume of a cylinder

#3d#volume#cylinder

Cylinder Surface Area

mensuration

S = 2\pi r(r+h)

Total surface area of a cylinder

#3d#surface#cylinder

Cone Volume

mensuration

V = \frac{1}{3}\pi r^2 h

Volume of a cone

#3d#volume#cone

Cone Curved Surface Area

mensuration

S = \pi r l

Lateral surface area of a cone with slant height l

#3d#surface#cone

Cuboid Volume

mensuration

V = l \times w \times h

Volume of a cuboid

#3d#volume#cuboid

Hemisphere Volume

mensuration

V = \frac{2}{3}\pi r^3

Volume of a hemisphere

#3d#volume#hemisphere

GCD-LCM Relationship

numberTheory

\gcd(a,b) \times \text{lcm}(a,b) = |ab|

Product of GCD and LCM equals product of numbers

#gcd#lcm

Fermat's Little Theorem

numberTheory

a^{p-1} \equiv 1 \pmod{p}

For prime p and integer a not divisible by p

#modular#prime#fermat

Euler's Totient Theorem

numberTheory

a^{\phi(n)} \equiv 1 \pmod{n}

Generalization of Fermat's Little Theorem

#modular#euler#totient

Wilson's Theorem

numberTheory

(p-1)! \equiv -1 \pmod{p}

A number p is prime iff this holds

#prime#factorial#wilson

Euclidean Algorithm

numberTheory

\gcd(a,b) = \gcd(b, a \bmod b)

Recursive algorithm for finding GCD

#gcd#algorithm#euclidean

Divisibility Rule for 3

numberTheory

n \equiv 0 \pmod{3} \iff \text{digit sum} \equiv 0 \pmod{3}

A number is divisible by 3 if its digit sum is

#divisibility#rules

Modular Addition

numberTheory

(a + b) \bmod n = ((a \bmod n) + (b \bmod n)) \bmod n

Modular arithmetic addition property

#modular#addition

Modular Multiplication

numberTheory

(a \cdot b) \bmod n = ((a \bmod n) \cdot (b \bmod n)) \bmod n

Modular arithmetic multiplication property

#modular#multiplication

Chinese Remainder Theorem

numberTheory

x \equiv a_i \pmod{n_i} \text{ has unique solution mod } N = \prod n_i

System of congruences with coprime moduli

#crt#congruence

Prime Counting (Approx)

numberTheory

\pi(x) \sim \frac{x}{\ln x}

Approximate number of primes up to x

#prime#counting#approximation

De Morgan's Law (Union)

setsAndFunctions

(A \cup B)^c = A^c \cap B^c

Complement of a union is intersection of complements

#sets#complement#de morgan

De Morgan's Law (Intersection)

setsAndFunctions

(A \cap B)^c = A^c \cup B^c

Complement of an intersection is union of complements

#sets#complement#de morgan

Inclusion-Exclusion (2 Sets)

setsAndFunctions

|A \cup B| = |A| + |B| - |A \cap B|

Cardinality of union of two sets

#sets#cardinality#counting

Inclusion-Exclusion (3 Sets)

setsAndFunctions

|A \cup B \cup C| = |A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B\cap C|

Cardinality of union of three sets

#sets#cardinality#counting

Function Composition

setsAndFunctions

(f \circ g)(x) = f(g(x))

Applying function f to result of function g

#functions#composition

Inverse Function

setsAndFunctions

f(f^{-1}(x)) = x = f^{-1}(f(x))

Property of an inverse function

#functions#inverse

Cartesian Product

setsAndFunctions

A \times B = \{(a,b) \mid a \in A, b \in B\}

Set of all ordered pairs from A and B

#sets#product#cartesian

Power Set Cardinality

setsAndFunctions

|\mathcal{P}(A)| = 2^{|A|}

Number of subsets of a set with n elements

#sets#power set#cardinality

Set Difference

setsAndFunctions

A \setminus B = A \cap B^c

Elements in A but not in B

#sets#difference

Symmetric Difference

setsAndFunctions

A \triangle B = (A \setminus B) \cup (B \setminus A)

Elements in exactly one of the two sets

#sets#symmetric#difference

Categories

Quadratic Formula

Difference of Squares

Sum of Cubes

Difference of Cubes

Binomial Theorem

Arithmetic Progression (nth Term)

Arithmetic Progression (Sum)

Geometric Progression (nth Term)

Geometric Progression (Sum)

Infinite Geometric Series

Vieta's Formulas (Quadratic)

Completing the Square

Pythagorean Identity

Secant & Tangent Identity

Cosecant & Cotangent Identity

Sine Addition & Subtraction

Cosine Addition & Subtraction

Tangent Addition & Subtraction

Double Angle (Sine)

Double Angle (Cosine)

Half Angle Formulas

Law of Sines

Law of Cosines

Power Rule (Derivative)

Product Rule

Quotient Rule

Chain Rule

Derivative of Exponential

Derivative of Logarithm

Standard Trig Derivatives

Power Rule (Integral)

Reciprocal Integral

Integration by Parts

Fundamental Theorem of Calculus

L'Hôpital's Rule

Product Rule for Logs

Quotient Rule for Logs

Power Rule for Logs

Change of Base Formula

Identity & Zero Properties

Inverse Properties

Natural Log & Euler Constant

Exponential Laws

Permutations Formula

Combinations Formula

Addition Rule of Probability

Multiplication Rule

Conditional Probability

Bayes' Theorem

Binomial Distribution PMF

Poisson Distribution PMF

Expected Value (Discrete)

Variance & Standard Deviation

Standard Normal Distribution (PDF)

Distance Formula (2D)

Midpoint Formula

Slope of a Line

Slope-Intercept Form

Point-Slope Form

Standard Circle Equation

General Parabola Equation

Standard Ellipse Equation

Standard Hyperbola Equation

Section Formula (Internal)

Perpendicular Distance from Point to Line

Vector Dot Product

Vector Cross Product

Vector Magnitude

Unit Vector

Scalar Projection of A onto B

Matrix Determinant (2x2)

Matrix Inverse (2x2)

Characteristic Equation (Eigenvalues)

Euler's Formula

De Moivre's Theorem

Modulus of Complex Number

Complex Conjugate

Polar Form

Argument of Complex Number