Math Formula Library

33 essential formulas, each with what every variable means, why it works, an example and a calculator to try it.

Algebra

Difference of Squares

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

Factors the difference of two perfect squares.

Exponent Rules

a^{m} a^{n} = a^{m + n},; \frac{a ^{m}}{a ^{n}} = a^{m - n},; (a^{m})^{n} = a^{mn},; a^{- n} = \frac{1}{a ^{n}}

The laws for multiplying, dividing and raising powers.

Logarithm Rules

lo g_{b} (x y) = lo g_{b} x + lo g_{b} y,; lo g_{b} \frac{x}{y} = lo g_{b} x - lo g_{b} y,; lo g_{b} x^{k} = k lo g_{b} x

Product, quotient, power and change-of-base rules for logarithms.

Quadratic Formula

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

Solves any quadratic equation ax² + bx + c = 0.

Slope Formula

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

The slope (rate of change) of the line through two points.

Sum and Difference of Cubes

a^{3} \pm b^{3} = (a \pm b) (a^{2} \mp ab + b^{2})

Factors sums and differences of perfect cubes.

Vertex of a Parabola

h = - \frac{b}{2 a}, k = f (h)

Finds the turning point (vertex) of y = ax² + bx + c.

Calculus

Chain Rule

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

Derivative of a composite function.

Fundamental Theorem of Calculus

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

Connects definite integrals to antiderivatives.

Integration by Parts

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

Integrates products of functions; the reverse of the product rule.

L'Hôpital's Rule

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

Evaluates 0/0 and ∞/∞ indeterminate limits.

Power Rule (Derivatives)

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

Derivative of a power of x.

Product Rule

(f g)^{'} = f^{'} g + f g^{'}

Derivative of a product of two functions.

Quotient Rule

(\frac{f}{g})^{'} = \frac{f ^{'} g - f g ^{'}}{g ^{2}}

Derivative of a quotient.

Geometry

Area of a Circle

A = π r^{2}

Area enclosed by a circle of radius r.

Area of a Triangle

A = \frac{1}{2} bh

Half of base times height.

Distance Formula

d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

Distance between two points in the plane.

Heron's Formula

A = s (s - a) (s - b) (s - c), s = \frac{a + b + c}{2}

Triangle area from its three side lengths.

Midpoint Formula

M = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})

The point halfway between two points.

Pythagorean Theorem

a^{2} + b^{2} = c^{2}

Relates the sides of a right triangle.

Volume of a Cylinder

V = π r^{2} h

Base area times height.

Volume of a Sphere

V = \frac{4}{3} π r^{3}

Volume of a ball of radius r; surface area is 4πr².

Linear Algebra

Cramer's Rule

x_{i} = \frac{det ( A _{i} )}{det ( A )}

Solves square linear systems with determinants.

Determinant of a Matrix

det [a c b d] = a d - b c

Scalar that tells whether a matrix is invertible.

Inverse of a 2×2 Matrix

[a c b d]^{- 1} = \frac{1}{a d - b c} [d - c - b a]

Swap the diagonal, negate the off-diagonal, divide by the determinant.

Statistics

Binomial Probability

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

Probability of exactly k successes in n independent trials.

Combinations and Permutations

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

Counting selections (order irrelevant) and arrangements (order matters).

Mean (Average)

x ˉ = \frac{1}{n} i = 1 \sum n x_{i}

The arithmetic average of a data set.

Probability Rules

P (A \cup B) = P (A) + P (B) - P (A \cap B), P (A^{'}) = 1 - P (A)

Addition, complement and multiplication rules.

Standard Deviation

s = \frac{\sum ( x _{i} - x ˉ ) ^{2}}{n - 1}

Measures spread of data around the mean.

Trigonometry

Law of Cosines

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

Generalizes the Pythagorean theorem to all triangles.

Law of Sines

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

Relates sides to the sines of their opposite angles in any triangle.

Pythagorean Identity

sin^{2} θ + cos^{2} θ = 1

The fundamental trigonometric identity.