[ { "Name": "Quadratic Formula", "Category": "Algebra", "Latex": "x=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}", "Description": "Finds the roots of a quadratic equation ax\u00B2 \u002B bx \u002B c = 0 when a is nonzero.", "Variables": [ "a: coefficient of x\u00B2 (a \u2260 0)", "b: coefficient of x", "c: constant term", "x: solution or root" ], "Assumptions": [ "The discriminant b\u00B2 \u2212 4ac determines the number and type of real roots.", "For the Quadratic Formula, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For a real-valued result, each even-root radicand must be nonnegative unless complex values are intended.", "The plus-minus sign represents separate branches; evaluate and verify both when both are admissible." ], "url": "/formula/quadratic-formula/" }, { "Name": "Pythagorean Theorem", "Category": "Geometry", "Latex": "a^2\u002Bb^2=c^2", "Description": "Relates the side lengths of a right triangle.", "Variables": [ "a, b: perpendicular leg lengths", "c: hypotenuse length" ], "Assumptions": [ "The theorem applies only to right triangles.", "Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula." ], "url": "/formula/pythagorean-theorem/" }, { "Name": "Slope-Intercept Form", "Category": "Algebra", "Latex": "y=mx\u002Bb", "Description": "Expresses a straight line using its slope and vertical intercept.", "Variables": [ "m: slope of the line", "b: y-intercept", "x: horizontal coordinate", "y: vertical coordinate" ], "Assumptions": [ "A positive slope rises from left to right; a negative slope falls.", "For the Slope-Intercept Form, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "url": "/formula/slope-intercept-form/" }, { "Name": "Distance Formula", "Category": "Algebra", "Latex": "d=\\sqrt{(x_2-x_1)^2\u002B(y_2-y_1)^2}", "Description": "Calculates the straight-line distance between two points in a coordinate plane.", "Variables": [ "(x\u2081, y\u2081): first point", "(x\u2082, y\u2082): second point", "d: distance between the points" ], "Assumptions": [ "The order of the two points does not change the result.", "For a real-valued result, each even-root radicand must be nonnegative unless complex values are intended.", "For the Distance Formula, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "url": "/formula/distance-formula/" }, { "Name": "Area of a Circle", "Category": "Geometry", "Latex": "A=\\pi r^2", "Description": "Calculates the area enclosed by a circle of radius r.", "Variables": [ "A: area", "r: radius", "\u03C0: circle constant, approximately 3.14159" ], "Assumptions": [ "The radius is half the diameter.", "Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula." ], "url": "/formula/circle-area/" }, { "Name": "Circumference of a Circle", "Category": "Geometry", "Latex": "C=2\\pi r=\\pi d", "Description": "Calculates the distance around a circle from its radius or diameter.", "Variables": [ "C: circumference", "r: radius", "d: diameter", "\u03C0: circle constant" ], "Assumptions": [ "The diameter equals 2r.", "Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula." ], "url": "/formula/circle-circumference/" }, { "Name": "Area of a Triangle", "Category": "Geometry", "Latex": "A=\\frac{1}{2}bh", "Description": "Calculates triangle area from a base and its perpendicular height.", "Variables": [ "A: area", "b: chosen base length", "h: perpendicular height to that base" ], "Assumptions": [ "The height must be perpendicular to the base.", "For the Area of a Triangle, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula." ], "url": "/formula/triangle-area/" }, { "Name": "Volume of a Sphere", "Category": "Geometry", "Latex": "V=\\frac{4}{3}\\pi r^3", "Description": "Calculates the volume enclosed by a sphere.", "Variables": [ "V: volume", "r: sphere radius", "\u03C0: circle constant" ], "Assumptions": [ "Volume is measured in cubic units.", "For the Volume of a Sphere, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula." ], "url": "/formula/sphere-volume/" }, { "Name": "Compound Interest Formula", "Category": "Applied Math", "Latex": "A=P\\left(1\u002B\\frac{r}{n}\\right)^{nt}", "Description": "Calculates a balance with interest compounded a fixed number of times per year.", "Variables": [ "A: final amount", "P: principal", "r: annual rate as a decimal", "n: compounding periods per year", "t: years" ], "Assumptions": [ "The stated rate must be converted from percent to decimal.", "For the Compound Interest Formula, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Map every symbol to the modeled quantity and verify units, domain restrictions, and simplifying assumptions." ], "url": "/formula/compound-interest/" }, { "Name": "Percentage Change Formula", "Category": "Applied Math", "Latex": "\\%\\text{ change}=\\frac{\\text{new}-\\text{old}}{|\\text{old}|}\\times100\\%", "Description": "Measures relative increase or decrease from an original nonzero value.", "Variables": [ "old: original value", "new: later value", "% change: signed relative difference" ], "Assumptions": [ "The original value must be nonzero.", "For the Percentage Change Formula, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Map every symbol to the modeled quantity and verify units, domain restrictions, and simplifying assumptions." ], "url": "/formula/percentage-change/" }, { "Name": "Arithmetic Sequence Formula", "Category": "Algebra", "Latex": "a_n=a_1\u002B(n-1)d", "Description": "Finds the nth term of a sequence with a constant difference.", "Variables": [ "a\u2099: nth term", "a\u2081: first term", "n: term number", "d: common difference" ], "Assumptions": [ "The difference between consecutive terms must be constant.", "For the Arithmetic Sequence Formula, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "url": "/formula/arithmetic-sequence/" }, { "Name": "Geometric Sequence Formula", "Category": "Algebra", "Latex": "a_n=a_1r^{n-1}", "Description": "Finds the nth term of a sequence with a constant ratio.", "Variables": [ "a\u2099: nth term", "a\u2081: first term", "r: common ratio", "n: term number" ], "Assumptions": [ "Each term is obtained by multiplying the previous term by r.", "For the Geometric Sequence Formula, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "url": "/formula/geometric-sequence/" }, { "Name": "Binomial Theorem", "Category": "Algebra", "Latex": "(a\u002Bb)^n=\\sum_{k=0}^{n}\\binom{n}{k}a^{n-k}b^k", "Description": "Expands a nonnegative integer power of a binomial.", "Variables": [ "a, b: binomial terms", "n: nonnegative integer exponent", "k: summation index", "C(n,k): binomial coefficient" ], "Assumptions": [ "The coefficients are the entries in row n of Pascal\u2019s triangle.", "For the Binomial Theorem, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "For the Binomial Theorem, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "url": "/formula/binomial-theorem/" }, { "Name": "Chain Rule", "Category": "Calculus", "Latex": "\\frac{d}{dx}f(g(x))=f\u0027(g(x))g\u0027(x)", "Description": "Differentiates a composite function by combining outer and inner derivatives.", "Variables": [ "f: outer function", "g: inner function", "x: independent variable" ], "Assumptions": [ "Differentiate the outer function while leaving the inner expression in place, then multiply by the inner derivative.", "For the Chain Rule, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "The function must satisfy the differentiability, continuity, or integrability conditions required by the operation being used." ], "url": "/formula/chain-rule/" }, { "Name": "Product Rule", "Category": "Calculus", "Latex": "(fg)\u0027=f\u0027g\u002Bfg\u0027", "Description": "Differentiates the product of two differentiable functions.", "Variables": [ "f, g: differentiable functions", "f\u2032, g\u2032: their derivatives" ], "Assumptions": [ "Differentiate one factor at a time while keeping the other factor unchanged.", "The function must satisfy the differentiability, continuity, or integrability conditions required by the operation being used." ], "url": "/formula/product-rule/" }, { "Name": "Integration by Parts", "Category": "Calculus", "Latex": "\\int u\\,dv=uv-\\int v\\,du", "Description": "Transforms an integral of a product using the product rule in reverse.", "Variables": [ "u: selected function to differentiate", "dv: selected differential to integrate", "du: derivative of u", "v: antiderivative of dv" ], "Assumptions": [ "Choose u so that differentiating it simplifies the expression.", "Integration limits, the differential, and any constant of integration must be included where required.", "The function must satisfy the differentiability, continuity, or integrability conditions required by the operation being used." ], "url": "/formula/integration-by-parts/" }, { "Name": "Derivative Definition", "Category": "Calculus", "Latex": "f\u0027(x)=\\lim_{h\\to0}\\frac{f(x\u002Bh)-f(x)}{h}", "Description": "Defines the instantaneous rate of change of a function as a limit.", "Variables": [ "f(x): original function", "h: change in the input", "f\u2032(x): derivative at x" ], "Assumptions": [ "The limit must exist for the derivative to exist.", "For the Derivative Definition, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "The function must satisfy the differentiability, continuity, or integrability conditions required by the operation being used." ], "url": "/formula/derivative-definition/" }, { "Name": "Bayes\u2019 Theorem", "Category": "Probability", "Latex": "P(A\\mid B)=\\frac{P(B\\mid A)P(A)}{P(B)}", "Description": "Updates the probability of an event after observing evidence.", "Variables": [ "P(A|B): posterior probability", "P(B|A): likelihood", "P(A): prior probability", "P(B): evidence probability" ], "Assumptions": [ "P(B) must be greater than zero.", "For the Bayes\u2019 Theorem, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "The event model, conditioning information, independence assumptions, and probability range must match the problem." ], "url": "/formula/bayes-theorem/" }, { "Name": "Conditional Probability", "Category": "Probability", "Latex": "P(A\\mid B)=\\frac{P(A\\cap B)}{P(B)}", "Description": "Calculates the probability of A when B is known to have occurred.", "Variables": [ "P(A|B): probability of A given B", "P(A \u2229 B): probability both occur", "P(B): probability of the condition" ], "Assumptions": [ "P(B) must be nonzero.", "For the Conditional Probability, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "The event model, conditioning information, independence assumptions, and probability range must match the problem." ], "url": "/formula/conditional-probability/" }, { "Name": "Sample Mean", "Category": "Statistics", "Latex": "\\bar{x}=\\frac{1}{n}\\sum_{i=1}^{n}x_i", "Description": "Calculates the arithmetic average of n observed values.", "Variables": [ "x\u0304: sample mean", "x\u1D62: ith observation", "n: number of observations" ], "Assumptions": [ "Every observation contributes equally.", "For the Sample Mean, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Sample Mean, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "For the Sample Mean, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions." ], "url": "/formula/sample-mean/" }, { "Name": "Sample Variance", "Category": "Statistics", "Latex": "s^2=\\frac{1}{n-1}\\sum_{i=1}^{n}(x_i-\\bar{x})^2", "Description": "Measures sample spread using squared deviations and Bessel\u2019s correction.", "Variables": [ "s\u00B2: sample variance", "x\u1D62: observation", "x\u0304: sample mean", "n: sample size" ], "Assumptions": [ "The denominator n \u2212 1 is used when estimating population variance from a sample.", "For the Sample Variance, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Sample Variance, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "For the Sample Variance, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions." ], "url": "/formula/sample-variance/" }, { "Name": "Z-Score Formula", "Category": "Statistics", "Latex": "z=\\frac{x-\\mu}{\\sigma}", "Description": "Expresses how many population standard deviations a value lies from the mean.", "Variables": [ "z: standardized score", "x: observed value", "\u03BC: population mean", "\u03C3: population standard deviation" ], "Assumptions": [ "A positive z-score lies above the mean; a negative one lies below.", "For the Z-Score Formula, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Z-Score Formula, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions." ], "url": "/formula/z-score/" }, { "Name": "Vector Dot Product", "Category": "Linear Algebra", "Latex": "\\mathbf{a}\\cdot\\mathbf{b}=\\sum_{i=1}^{n}a_ib_i=\\|\\mathbf{a}\\|\\|\\mathbf{b}\\|\\cos\\theta", "Description": "Combines two vectors into a scalar and relates their components to the angle between them.", "Variables": [ "a, b: vectors", "a\u1D62, b\u1D62: vector components", "\u03B8: angle between vectors" ], "Assumptions": [ "Perpendicular nonzero vectors have dot product zero.", "For the Vector Dot Product, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "Preserve matrix order and verify dimension compatibility; multiplication and inversion are not generally commutative or always defined.", "The angle unit and the domain or branch of any inverse trigonometric function must be explicit." ], "url": "/formula/dot-product/" }, { "Name": "Cross Product Components", "Category": "Linear Algebra", "Latex": "\\mathbf a\\times\\mathbf b=\\begin{bmatrix}a_2b_3-a_3b_2\\\\a_3b_1-a_1b_3\\\\a_1b_2-a_2b_1\\end{bmatrix}", "Description": "Calculates the three-dimensional vector cross product.", "Variables": [ "a,b: 3D vectors", "result: the quantity produced after the stated inputs and conditions are applied" ], "Assumptions": [ "The result is perpendicular to both vectors.", "Preserve matrix order and verify dimension compatibility; multiplication and inversion are not generally commutative or always defined.", "Matrix and vector dimensions must be compatible, and any required inverse, determinant, rank, or basis condition must hold." ], "url": "/formula/cross-product-formula/" }, { "Name": "2\u00D72 Matrix Determinant", "Category": "Linear Algebra", "Latex": "\\det\\begin{pmatrix}a\u0026b\\\\c\u0026d\\end{pmatrix}=ad-bc", "Description": "Calculates the determinant of a two-by-two matrix.", "Variables": [ "a, b, c, d: matrix entries", "det: scalar determinant" ], "Assumptions": [ "A zero determinant means the matrix is singular and has no inverse.", "Preserve matrix order and verify dimension compatibility; multiplication and inversion are not generally commutative or always defined.", "Matrix and vector dimensions must be compatible, and any required inverse, determinant, rank, or basis condition must hold." ], "url": "/formula/two-by-two-determinant/" } ]