Equations Compendium

Quadratic Formula

x = −b ± √(b²−4ac)2a

Solves ax² + bx + c = 0 for any real coefficients.

Difference of Squares

a² − b² = (a+b)(a−b)

Factoring pattern used constantly in simplification.

Perfect Square Trinomial

(a±b)² = a² ± 2ab + b²

The square of a binomial, expanded.

Sum / Difference of Cubes

a³±b³ = (a±b)(a²∓ab+b²)

Higher-order factoring pattern.

Slope of a Line

m = y₂−y₁x₂−x₁

Rise over run between two points.

Slope-Intercept Form

y = mx + b

m is slope, b is the y-intercept.

Point-Slope Form

y − y₁ = m(x − x₁)

A line through one known point with slope m.

Distance Formula

d = √((x₂−x₁)²+(y₂−y₁)²)

The Pythagorean theorem, restated for coordinates.

Midpoint Formula

M = x₁+x₂2, y₁+y₂2

The point exactly between two coordinates.

Laws of Exponents

aᵐ·aⁿ = aᵐ⁺ⁿ   (aᵐ)ⁿ = aᵐⁿ

Multiplying and raising powers.

Laws of Logarithms

log(ab) = log a + log b

Logs turn multiplication into addition.

Change of Base

log_bx = log xlog b

Convert any log to a base your calculator has.

Pythagorean Theorem

a² + b² = c²

Right-triangle legs and hypotenuse.

Triangle Area

A = 12bh

Half the base times the height.

Heron's Formula

A = √(s(s−a)(s−b)(s−c))

Triangle area from three side lengths; s is the semiperimeter.

Circle Circumference

C = 2πr

Distance around a circle.

Circle Area

A = πr²

Space enclosed by a circle.

Rectangle / Parallelogram Area

A = bh

Base times height (perpendicular).

Trapezoid Area

A = 12(b₁+b₂)h

Average of the two parallel sides, times height.

Regular Polygon Interior Angle

θ = (n−2)·180°n

For an n-sided regular polygon.

Sphere Volume

V = 43πr³

Volume enclosed by a sphere.

Sphere Surface Area

SA = 4πr²

Surface area of a sphere.

Cylinder Volume

V = πr²h

Circle area times height.

Cone Volume

V = 13πr²h

One-third of the equivalent cylinder.

Rectangular Prism Volume

V = lwh

Length × width × height.

Pyramid Volume

V = 13Bh

One-third base area times height.

SOH-CAH-TOA

sin=opphyp cos=adjhyp tan=oppadj

The three basic ratios in a right triangle.

Pythagorean Identity

sin²θ + cos²θ = 1

The most-used trig identity.

Law of Sines

asin A = bsin B = csin C

Relates sides to opposite angles in any triangle.

Law of Cosines

c² = a²+b²−2ab·cos C

Generalized Pythagorean theorem for any triangle.

Angle Sum Formulas

sin(A±B) = sinA cosB ± cosA sinB

Expanding sine of a sum or difference.

Double Angle

sin2θ = 2sinθcosθ

Sine of double an angle.

Radians ↔ Degrees

rad = deg × π180

Converting between angle units.

Unit Circle Point

(cosθ, sinθ)

Coordinates of angle θ on the unit circle.

Power Rule (derivative)

d/dx[xⁿ] = nxⁿ⁻¹

The single most-used differentiation rule.

Product Rule

(fg)′ = f′g + fg′

Derivative of a product of two functions.

Quotient Rule

(f/g)′ = f′g − fg′g²

Derivative of a quotient.

Chain Rule

d/dx[f(g(x))] = f′(g(x))·g′(x)

Derivative of a composed function.

Derivative of eˣ

d/dx[eˣ] = eˣ

The function that is its own derivative.

Derivative of ln x

d/dx[ln x] = 1/x

Derivative of the natural log.

Power Rule (integral)

∫xⁿdx = xⁿ⁺¹n+1 + C

Antiderivative of a power, n ≠ −1.

Fundamental Theorem of Calculus

∫_a^b f′(x)dx = f(b) − f(a)

Links derivatives and definite integrals.

Limit Definition of a Derivative

f′(x) = lim_h→0 f(x+h)−f(x)h

Where the derivative comes from.

Arithmetic Sequence

aₙ = a₁ + (n−1)d

Each term adds a constant difference d.

Arithmetic Series Sum

Sₙ = n2(a₁+aₙ)

Sum of the first n terms.

Geometric Sequence

aₙ = a₁·rⁿ⁻¹

Each term multiplies by a constant ratio r.

Geometric Series Sum

Sₙ = a₁1−rⁿ1−r

Sum of the first n geometric terms, r ≠ 1.

Infinite Geometric Sum

S = a₁1−r

Converges only when |r| < 1.

Fibonacci Recurrence

Fₙ = Fₙ₋₁ + Fₙ₋₂

Each term is the sum of the two before it.

Mean

x̄ = Σxn

Sum of values divided by count.

Standard Deviation

σ = √(Σ(x−x̄)²n)

Typical distance of a value from the mean.

Combinations

ₙCᵣ = n!r!(n−r)!

Ways to choose r items from n, order not mattering.

Permutations

ₙPᵣ = n!(n−r)!

Ways to arrange r items from n, order mattering.

Probability of A and B (independent)

P(A∩B) = P(A)·P(B)

Both independent events occurring.

Probability of A or B

P(A∪B) = P(A)+P(B)−P(A∩B)

Either event occurring; subtract the overlap.