Mathematics

Algebra and finance formulas, explained with worked examples.

This section covers the math formulas students run into most often between middle school and the first year or two of college: solving equations (the quadratic formula, slope-intercept form), coordinate geometry (distance and midpoint), calculus fundamentals (derivative and integral rules), trigonometry (the unit circle relationships and the laws of sines and cosines), basic statistics (mean, median, mode, standard deviation), and the compound-interest math that shows up in both algebra classes and personal finance.

Each page is built the same way on purpose: a plain-language explanation of what the formula means and why it works, the variables spelled out individually, at least one fully worked example with every intermediate step shown (not just the final answer), a free calculator so you can check your own work, and a short list of mistakes students commonly make with that specific formula. If you're prepping for an exam, start with the subcategory that matches your course — Algebra for equation-solving, Calculus for derivatives and integrals, Trigonometry for triangles and periodic functions, Statistics for data analysis, or Financial Math for interest calculations.

Algebra

x = (-b ± √(b² - 4ac)) / 2a

Quadratic Formula

Find the roots of any quadratic equation ax² + bx + c = 0, even when it can't be factored by inspection.

a² + b² = c²

Pythagorean Theorem

Relates the three sides of a right triangle, letting you find any one side from the other two.

Slope-Intercept Form

The most common way to write the equation of a straight line, revealing its slope and y-intercept directly.

d = √((x₂-x₁)² + (y₂-y₁)²)

Distance Formula

Calculates the straight-line distance between two points on a coordinate plane.

M = ((x₁+x₂)/2, (y₁+y₂)/2)

Midpoint Formula

Finds the exact point halfway between two coordinates by averaging their x- and y-values.

Finance

SI = (P × R × T) / 100

Simple Interest

Calculate interest earned on a principal amount at a fixed annual rate.

A = P(1 + r/n)^(nt)

Compound Interest

Calculate how an investment or loan grows when interest is calculated repeatedly on both the principal and previously earned interest.

Calculus — Derivatives

Calculus — Derivatives

d/dx[xⁿ] = nxⁿ⁻¹

Power Rule

The most fundamental differentiation rule — multiply by the exponent, then reduce the exponent by one.

Calculus — Derivatives

d/dx(fg) = fg' + gf'

Product Rule

Differentiates a product of two functions — you cannot simply multiply their individual derivatives.

Calculus — Derivatives

d/dx(f/g) = (gf' − fg') / g²

Quotient Rule

Differentiates a ratio of two functions — remembered by the mnemonic "low d-high minus high d-low, over low squared."

Calculus — Derivatives

dy/dx = (dy/du) · (du/dx)

Chain Rule

Differentiates composite functions — functions nested inside other functions.

Calculus — Derivatives

d/dx(sin x) = cos x

Trigonometric Derivatives

The derivatives of all six trigonometric functions — sin, cos, tan, cot, sec, and csc.

Calculus — Derivatives

d/dx(arcsin x) = 1/√(1−x²)

Inverse Trig Derivatives

The derivatives of the inverse trigonometric functions — arcsin, arccos, arctan, and arccot.

Calculus — Derivatives

d/dx(eˣ) = eˣ

Exponential & Log Derivatives

The three core derivatives involving exponential and logarithmic functions: eˣ, aˣ, and ln x.

Calculus — Integrals

Calculus — Integrals

∫ eˣ dx = eˣ + C

Basic Integration Rules

The essential integral formulas for constants, 1/x, eˣ, aˣ, and ln x.

Calculus — Integrals

∫ sin x dx = −cos x + C

Trigonometric Integrals

The complete table of ten standard trigonometric integrals — each the reverse of a derivative rule.

Calculus — Integrals

∫ dx/√(a²−x²) = arcsin(x/a) + C

Special Integrals

Two integral forms that arise from square roots of quadratic expressions, producing inverse-trig or logarithmic results.

Trigonometry

sin θ = O/H, cos θ = A/H, tan θ = O/A

Sine, Cosine & Tangent

The three fundamental trigonometric ratios for right triangles, remembered with SOH-CAH-TOA.

a/sin(A) = b/sin(B) = c/sin(C)

Law of Sines

Relates the sides of any triangle to the sines of their opposite angles — not just right triangles.

c² = a² + b² − 2ab·cos(C)

Law of Cosines

A generalization of the Pythagorean theorem that works for any triangle, not just right triangles.

Statistics

x̄ = (Σx) / n

Mean, Median & Mode

The three measures of central tendency for a data set — mean (average), median (middle), and mode (most frequent).

σ = √(Σ(x−μ)² / n)

Standard Deviation

Measures how spread out data values are from the mean.

C(n,r) = n! / (r!(n−r)!)

Combinations & Permutations

Count the number of ways to choose (combinations) or arrange (permutations) items from a group.