Formula Library

Every formula you need, explained simply.

Quadratic Formula

Algebra I

x = (−b ± √(b² − 4ac)) / 2a

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

Slope Formula

Algebra I

m = (y₂ − y₁) / (x₂ − x₁)

The steepness of a line between two points.

Slope-Intercept Form

Algebra I

y = mx + b

A line where m is slope and b is the y-intercept.

Point-Slope Form

Algebra I

y − y₁ = m(x − x₁)

A line through point (x₁, y₁) with slope m.

Distance Formula

Geometry

d = √((x₂ − x₁)² + (y₂ − y₁)²)

Distance between two points on a coordinate plane.

Pythagorean Theorem

Geometry

a² + b² = c²

Relates the sides of a right triangle.

Area of a Circle

Geometry

A = πr²

Area enclosed by a circle of radius r.

Circumference

Geometry

C = 2πr

Perimeter (distance around) a circle.

Area of a Triangle

Geometry

A = ½bh

Area with base b and height h.

Circle Equation

Geometry

(x − h)² + (y − k)² = r²

A circle centered at (h, k) with radius r.

Law of Sines

Precalculus

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

Relates sides and angles of any triangle.

Law of Cosines

Precalculus

c² = a² + b² − 2ab·cos C

Generalized Pythagorean theorem for any triangle.

Pythagorean Identity

Precalculus

sin²(θ) + cos²(θ) = 1

Fundamental trigonometric identity.

Double Angle (Sine)

Precalculus

sin(2θ) = 2·sin(θ)·cos(θ)

Expresses sin of double an angle.

Power Rule (Derivative)

Calculus

d/dx [xⁿ] = n·xⁿ⁻¹

The most-used derivative rule for polynomials.

Product Rule (Derivative)

Calculus

(fg)′ = f′g + fg′

Derivative of a product of two functions.

Quotient Rule (Derivative)

Calculus

(f/g)′ = (f′g − fg′) / g²

Derivative of a quotient of two functions.

Chain Rule (Derivative)

Calculus

d/dx [f(g(x))] = f′(g(x))·g′(x)

Derivative of a composite function.

Integral of xⁿ

Calculus

∫xⁿ dx = xⁿ⁺¹ / (n+1) + C

Basic power rule for integration (n ≠ −1).

Mean (Average)

Statistics

μ = Σx / N

Sum of all values divided by count.

Standard Deviation

Statistics

σ = √(Σ(x − μ)² / N)

Measures how spread out the data is.

Probability

Statistics

P(A) = favorable / total

Likelihood of an event occurring.

Combination

Statistics

C(n, r) = n! / (r!(n − r)!)

Number of ways to choose r items from n (order doesn't matter).

Compound Interest

Algebra II

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

Interest earned on both principal and accumulated interest.

Simple Interest

Pre-Algebra

I = P × r × t

Interest on principal only.

Percent Change

Basic Math

Δ% = (new − old) / old × 100

Percentage increase or decrease.

Factoring (Difference of Squares)

Algebra I

a² − b² = (a + b)(a − b)

Factors a difference of two squares.

Logarithm Rule

Algebra II

logₐ(xy) = logₐ(x) + logₐ(y)

Product rule for logarithms.

Exponential Growth

Precalculus

A = P·e^(rt)

Continuous compound growth.

Matrix Determinant (2×2)

College Math

det = ad − bc

Determinant of a 2×2 matrix [[a,b],[c,d]].

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