Master foundational physics, finance, calculus, and engineering equations with live evaluators, derivations, and worked examples.
Closed-form algebraic solution for the roots of any second-degree polynomial equation ax² + bx + c = 0, derived via completing the square.
Algebraic expansion of powers of a binomial (a + b)ⁿ into a sum of n + 1 terms weighted by combinatorial binomial coefficients C(n, k).
Represents a smooth differentiable function f(x) near center point x = a as an infinite power series constructed from its k-th derivatives.
Transforms the integral of a product of functions into an easier integral by inverting the differential product rule.
Unifies differential and integral calculus by proving that definite integration is the exact inverse operation of differentiation.
Connects complex exponential functions with trigonometric sine and cosine waves on the unit circle in the complex plane.
Computes the n-th term and exact cumulative sum of an arithmetic progression with constant step difference d.
Closed-form sum of n terms in a geometric progression with common multiplier ratio r, plus infinite convergence when |r| < 1.
Complete reference for logarithmic product, quotient, power, and change-of-base identities across natural (ln), binary (log₂), and common (log₁₀) bases.
Core operational differentiation rules for composite functions, products, quotients, and power functions in single-variable and multivariable calculus.
Generalizes the Pythagorean theorem to any oblique triangle by adding the −2ab cos(C) projection term.
States that the ratio of any side length of a triangle to the sine of its opposite interior angle is constant and equal to the circumcircle diameter 2R.
Computes the exact surface area, altitudes, inradius, and circumradius of any triangle directly from its three side lengths a, b, c without measuring angles.
Euclidean metric distance, segment midpoint, and gradient slope between two Cartesian points (x₁, y₁) and (x₂, y₂).
Analytic geometry equations for circles and ellipses including center (h, k), semi-major/minor axes (a, b), eccentricity e, focal points, area, and Ramanujan perimeter.
Fundamental orthogonal metric relating legs and hypotenuse in right triangles and rectangular 3D box diagonals.
Archimedes' solid geometry formulas comparing volumes and surface areas of spheres, right circular cylinders, and cones.
Relates circular radius r and central angle θ to arc length s, sector area, chord length, sagitta, and circular segment area.
Complete trigonometric reference for angle-sum, double-angle, half-angle, and power-reduction identities.
Computes the shortest great-circle surface distance between two latitude/longitude coordinates on a sphere of radius R (Earth mean R = 6,371 km).
Calculates accumulated future value A from principal P compounded n times per year at nominal annual interest rate r over t years.
Discounts a stream of n equal periodic cash flows PMT at periodic rate r to their lump-sum present value PV.
Nobel-winning partial differential equation closed-form solution for pricing European call and put options under geometric Brownian motion.
Links an asset's required expected return E(R_i) to the risk-free rate R_f, its non-diversifiable systematic beta β_i, and the equity market risk premium.
Blends a corporation's cost of equity R_e and after-tax cost of debt R_d(1 − T_c) weighted by market capital structure.
Decomposes Return on Equity (ROE) into three distinct operational drivers: profitability (Net Margin), capital efficiency (Asset Turnover), and financial leverage (Equity Multiplier).
Measures the net value added in today's dollars by discounting future project cash flows CF_t back to t = 0 at hurdle rate r and subtracting initial capex C₀.
Values a mature equity or terminal DCF cash flow stream growing perpetually at constant rate g < r.
Nobel laureate William Sharpe's reward-to-variability metric measuring portfolio excess return per unit of total volatility σ_p.
Bell Labs information-theoretic formula for the exact fraction of bankroll f* to allocate to a positive-expectancy wager or portfolio to maximize long-run compounding.
Connects net external force F_net on mass m to acceleration a and the four constant-acceleration SUVAT kinematic equations.
Inverse-square law governing gravitational attraction, surface gravity g, circular orbital velocity, escape velocity, and Kepler's Third Law.
States that in a conservative system free of non-conservative friction/drag, the sum of kinetic energy K and potential energy U remains invariant.
Relates the dot-product mechanical work W = F·d cos(θ) performed on a body to its exact change in kinetic energy ΔK and instantaneous power P = F·v.
Links force-time impulse J = F_avg Δt to momentum change Δp and governs elastic, partially elastic, and perfectly inelastic collisions.
Fundamental electrical circuit equations relating voltage V, current I, resistance R, Joule power P, and Kirchhoff's node/loop laws.
Inverse-square electrostatic force, electric field E, and electric potential V between point charges in vacuum or dielectric media.
Governs voltage generation in electric generators, transformers, inductors, and wireless chargers from time-varying magnetic flux Φ_B.
Unifies electricity, magnetism, and optics via Maxwell's four field equations, wave propagation speed c = fλ, and Planck photon energy E = hf.
Einstein's special relativity equations for the Lorentz factor γ, rest energy E₀ = mc², relativistic kinetic energy, time dilation, and length contraction.
Conservation of mechanical energy along a steady, incompressible, inviscid fluid streamline relating static pressure, dynamic velocity head, and elevation head.
Dimensionless ratio of inertial forces to viscous shear forces predicting laminar, transitional, or turbulent flow in pipes and boundary layers.
Universal hydraulic engineering equation calculating major frictional head loss and pressure drop along straight pressurized pipes.
Exact analytical solution of the Navier-Stokes equations for steady laminar volumetric flow through a cylindrical tube.
Structural mechanics formula determining the elastic axial compressive load at which a slender column undergoes sudden lateral buckling.
Fundamental solid mechanics relation linking internal bending moment M to longitudinal normal stress across a symmetric beam cross-section.
Calculates torsional shear stress and angle of twist in solid or hollow drive shafts transmitting mechanical torque.
Defines the −3 dB half-power corner frequency, phase shift, and exponential step-response time constant of first-order RC analog filters.
Relates absolute pressure, volume, molar quantity, and thermodynamic Kelvin temperature for dilute gases via universal gas constant R = 8.31446 J/(mol·K).
Quantifies the exponential sensitivity of chemical reaction rate constants k to activation energy barrier E_a and absolute temperature T.
Calculates the pH of aqueous weak-acid/conjugate-base buffer solutions and physiological bicarbonate equilibria.
Determines redox half-cell and battery open-circuit voltage away from standard 1 M conditions, as well as biological membrane equilibrium potentials.
Fundamental chemical thermodynamics criterion governing reaction spontaneity, maximum non-expansion work, and equilibrium constants at constant T and P.
Second-Law upper limit on thermal efficiency for any heat engine operating between hot reservoir T_H and cold sink T_C in Kelvin.
Analytical chemistry spectroscopy equation linking optical absorbance A linearly to molar absorptivity ε, cuvette path length l, and molar concentration c.
Updates prior belief P(A) into exact posterior probability P(A|B) upon observing diagnostic, experimental, or classification evidence B.
Canonical bell-curve probability density arising from the Central Limit Theorem, parameterized by mean μ and standard deviation σ.
Exact discrete probability of observing k successes in n independent Bernoulli trials each having success probability p.
Models the discrete count of independent arrivals or defects occurring in a fixed interval of time, area, or volume at average rate λ.
Measures dimensionless linear association r ∈ [−1, +1] and derives minimum-squared-error linear regression coefficients.
Quantifies sampling precision of a sample mean x̄ via standard error s/√n and constructs Frequentist confidence intervals.
Foundational information-theory equations quantifying minimum lossless compression bits per symbol H(X) and noisy communication channel capacity C.