Essential Math Formulas
Quick reference guide to the most important mathematical formulas organized by subject area.
📐 Algebra Formulas
Quadratic Equations
Quadratic Formula
$$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$$
For equations of the form $ax^2 + bx + c = 0$
Example: $x^2 - 5x + 6 = 0$ → $x = 2, 3$
Discriminant
$$D = b^2 - 4ac$$
Determines nature of roots: D > 0 (two real), D = 0 (one real), D < 0 (complex)
Vertex Form
$$y = a(x-h)^2 + k$$
Vertex at point $(h, k)$
Exponents & Logarithms
Exponent Rules
$a^m \cdot a^n = a^{m+n}$
$(a^m)^n = a^{mn}$
$\frac{a^m}{a^n} = a^{m-n}$
$a^0 = 1$ (when $a \neq 0$)
Logarithm Properties
$\log_a(xy) = \log_a x + \log_a y$
$\log_a(\frac{x}{y}) = \log_a x - \log_a y$
$\log_a(x^n) = n\log_a x$
$\log_a a = 1$
🔺 Geometry Formulas
Area Formulas
Triangle
$$A = \frac{1}{2}bh$$
Where $b$ = base, $h$ = height
$$A = \sqrt{s(s-a)(s-b)(s-c)}$$
Heron's formula: $s = \frac{a+b+c}{2}$
Circle
$$A = \pi r^2$$
$$C = 2\pi r$$
Where $r$ = radius, $C$ = circumference
Rectangle & Square
$$A_{rectangle} = lw$$
$$A_{square} = s^2$$
$l$ = length, $w$ = width, $s$ = side
Volume Formulas
Sphere
$$V = \frac{4}{3}\pi r^3$$
$$SA = 4\pi r^2$$
Cylinder
$$V = \pi r^2 h$$
$$SA = 2\pi r^2 + 2\pi rh$$
Cone
$$V = \frac{1}{3}\pi r^2 h$$
$$SA = \pi r^2 + \pi rl$$
$l$ = slant height
Coordinate Geometry
Distance Formula
$$d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$
Midpoint Formula
$$M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$$
Slope Formula
$$m = \frac{y_2-y_1}{x_2-x_1}$$
🌊 Trigonometry Formulas
Basic Trigonometric Functions
Basic Ratios
$\sin \theta = \frac{opposite}{hypotenuse}$
$\cos \theta = \frac{adjacent}{hypotenuse}$
$\tan \theta = \frac{opposite}{adjacent}$
Pythagorean Identity
$$\sin^2 \theta + \cos^2 \theta = 1$$
$$1 + \tan^2 \theta = \sec^2 \theta$$
$$1 + \cot^2 \theta = \csc^2 \theta$$
Law of Sines
$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$
Law of Cosines
$$c^2 = a^2 + b^2 - 2ab\cos C$$
Angle Addition Formulas
Sine Addition
$$\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$$
Cosine Addition
$$\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$$
Double Angle
$\sin 2\theta = 2\sin \theta \cos \theta$
$\cos 2\theta = \cos^2 \theta - \sin^2 \theta$
📈 Calculus Formulas
Derivatives
Power Rule
$$\frac{d}{dx}[x^n] = nx^{n-1}$$
Product Rule
$$\frac{d}{dx}[uv] = u'v + uv'$$
Quotient Rule
$$\frac{d}{dx}\left[\frac{u}{v}\right] = \frac{u'v - uv'}{v^2}$$
Chain Rule
$$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$$
Common Derivatives
Trigonometric
$\frac{d}{dx}[\sin x] = \cos x$
$\frac{d}{dx}[\cos x] = -\sin x$
$\frac{d}{dx}[\tan x] = \sec^2 x$
Exponential & Logarithmic
$\frac{d}{dx}[e^x] = e^x$
$\frac{d}{dx}[\ln x] = \frac{1}{x}$
$\frac{d}{dx}[a^x] = a^x \ln a$
Integration
Power Rule for Integration
$$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$
(when $n \neq -1$)
Fundamental Theorem
$$\int_a^b f(x)dx = F(b) - F(a)$$
where $F'(x) = f(x)$
📊 Statistics Formulas
Descriptive Statistics
Mean
$$\bar{x} = \frac{\sum x_i}{n}$$
Standard Deviation
$$s = \sqrt{\frac{\sum(x_i - \bar{x})^2}{n-1}}$$
Variance
$$s^2 = \frac{\sum(x_i - \bar{x})^2}{n-1}$$
Probability
Combination
$$C(n,r) = \frac{n!}{r!(n-r)!}$$
Permutation
$$P(n,r) = \frac{n!}{(n-r)!}$$
Normal Distribution
$$Z = \frac{x - \mu}{\sigma}$$
Z-score formula