Formula solver index — formula.expert

Distance Formula (2D)

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

GeometryAlgebraStraight-line distance between two points in the coordinate plane.

Slope Between Two Points

m = \frac{y_2 - y_1}{x_2 - x_1}

AlgebraGeometrySteepness of a line as rise over run between two points.

Slope-Intercept Form of a Line

y = mx + b

AlgebraGeometryEquation of a straight line from its slope and y-intercept.

Exponential Growth

A = A_0 (1 + r)^{t}

AlgebraAmount after t periods of compound growth at rate r per period.

Exponential Decay

A = A_0 (1 - r)^{t}

AlgebraAmount remaining after t periods of losing a fixed fraction r per period.

Exponential Growth by Doubling Time

N = N_0 \cdot 2^{t/T}

AlgebraGrowth of a quantity that doubles every fixed interval T.

Arithmetic Sequence nth Term

a_n = a_1 + (n - 1) d

AlgebraThe nth term of a sequence that steps by a fixed common difference.

Arithmetic Series Sum

S_n = \frac{n}{2} \left( 2 a_1 + (n - 1) d \right)

AlgebraSum of the first n terms of an arithmetic sequence.

Geometric Sequence nth Term

a_n = a_1 \, r^{\,n-1}

AlgebraThe nth term of a sequence that multiplies by a fixed ratio each step.

Geometric Series Sum

S_n = \frac{a_1 (1 - r^{n})}{1 - r}

AlgebraSum of the first n terms of a geometric sequence.

Infinite Geometric Series

S = \frac{a_1}{1 - r}

AlgebraLimit of a geometric series with ratio between −1 and 1, summed forever.

Simple Interest

I = P \, r \, t

AlgebraInterest earned on a principal at a flat rate per period, without compounding.

Compound Interest (Periodic)

A = P \left( 1 + \frac{r}{n} \right)^{n t}

AlgebraAmount after t periods when interest compounds n times per period at rate r.

Continuous Compounding

A = P \, e^{r t}

AlgebraAmount after t periods when interest compounds continuously at rate r.

Present Value

\mathit{PV} = \frac{\mathit{FV}}{(1 + r)^{t}}

AlgebraToday's value of a future amount, discounted at rate r per period.

Percent Change

c = \frac{x_{\text{new}} - x_{\text{old}}}{x_{\text{old}}}

AlgebraRelative change from an old value to a new one, as a fraction of the old value.

Distance Formula (3D)

d = \sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2}

AlgebraStraight-line distance between two points from their separations along each axis.

Quadratic Formula (Positive Root)

x = \frac{-b + \sqrt{b^2 - 4ac}}{2a}

AlgebraThe larger of the two solutions of ax² + bx + c = 0, taking the plus branch of the square root in the quadratic formula.

Quadratic Formula (Negative Root)

x = \frac{-b - \sqrt{b^2 - 4ac}}{2a}

AlgebraThe smaller of the two solutions of ax² + bx + c = 0, taking the minus branch of the square root in the quadratic formula.

Discriminant of a Quadratic

\Delta = b^2 - 4ac

AlgebraThe quantity b² − 4ac under the square root, which reveals how many real roots a quadratic has before you solve it.

Sum of the Roots of a Quadratic

S = -\frac{b}{a}

AlgebraVieta's relation giving the sum of a quadratic's two roots directly from its coefficients, without solving the equation.

Product of the Roots of a Quadratic

P = \frac{c}{a}

AlgebraVieta's relation giving the product of a quadratic's two roots directly from its coefficients, without solving the equation.

Vertex x-Coordinate of a Parabola

h = -\frac{b}{2a}

AlgebraGeometryThe x-coordinate of a parabola's vertex, sitting midway between the two roots and marking the axis of symmetry.

Vertex y-Coordinate of a Parabola

k = c - \frac{b^2}{4a}

AlgebraGeometryThe y-coordinate of a parabola's vertex — its minimum value when a is positive, and its maximum when a is negative.

Midpoint Formula

x_m = \frac{x_1 + x_2}{2}

AlgebraGeometryThe coordinate halfway between two endpoints along one axis; apply it to x and again to y for the midpoint of a segment.

Circle Radius from Centre and a Point

r = \sqrt{(x - h)^2 + (y - k)^2}

AlgebraGeometryRadius of a circle passing through a given point, measured from its centre using the distance formula inside the circle equation.

Distance from a Point to a Line

d = \frac{\left| A x_0 + B y_0 + C \right|}{\sqrt{A^2 + B^2}}

AlgebraGeometryPerpendicular distance from a point to a line written in general form Ax + By + C = 0, the shortest gap between the two.

Point-Slope Form of a Line

y = y_1 + m(x - x_1)

AlgebraGeometryEquation of a straight line through one known point with a known slope, giving the y-value at any x you care to choose.

Slope from Standard Form of a Line

m = -\frac{A}{B}

AlgebraGeometrySlope of a line written in standard form Ax + By = C, read straight off the two coefficients without rearranging anything.

x-Intercept of a Line

x_{\text{int}} = -\frac{b}{m}

AlgebraGeometryWhere a line in slope-intercept form crosses the x-axis, found by setting y to zero and solving the remaining equation for x.

Perpendicular Slope Relation

m_2 = -\frac{1}{m_1}

AlgebraGeometrySlope of a line perpendicular to a given line — the negative reciprocal, so that the two slopes always multiply to −1.

Logarithm Change of Base

\log_b x = \frac{\ln x}{\ln b}

AlgebraValue of a logarithm in any base, rewritten with natural logarithms so an ordinary calculator can evaluate it directly.

Logarithm of a Product

\log_b(xy) = \log_b x + \log_b y

AlgebraThe logarithm of a product equals the sum of the logarithms, the identity that once turned multiplication into simple addition.

Logarithm of a Quotient

\log_b\!\left(\frac{x}{y}\right) = \log_b x - \log_b y

AlgebraThe logarithm of a quotient equals the difference of the logarithms, turning a division problem into a simple subtraction.

Logarithm of a Power

\log_b\!\left(x^{n}\right) = n \log_b x

AlgebraThe logarithm of a power pulls the exponent out in front, the identity that unlocks any equation with the unknown in an exponent.

Solve an Exponential Equation for the Exponent

x = \frac{\ln (y / a)}{\ln b}

AlgebraSolves a·bˣ = y for the unknown exponent x using logarithms, the standard move behind every growth and decay problem.

Power of a Power

\left( b^{m} \right)^{n} = b^{mn}

AlgebraRaising a power to another power multiplies the two exponents, the exponent law behind every nested exponential expression.

nth Root as a Fractional Exponent

y = \sqrt[n]{x} = x^{1/n}

AlgebraThe nth root of a number written as a fractional exponent, so that roots and powers obey exactly the same set of rules.

Binomial Coefficient (n Choose k)

\binom{n}{k} = \frac{n!}{k! \, (n-k)!}

AlgebraThe number of ways to choose k items from n, which is also the coefficient of a single term in a binomial expansion.

Completing the Square: the Constant Needed

k = \left( \frac{b}{2} \right)^{2}

AlgebraThe constant you add to x² + bx to turn it into a perfect square trinomial, the pivotal step in completing the square.

Arithmetic Mean of Two Numbers

A = \frac{a + b}{2}

AlgebraThe ordinary average of two numbers, the balance point sitting exactly halfway between them on the number line.

Geometric Mean of Two Numbers

G = \sqrt{ab}

AlgebraGeometryThe square root of the product of two numbers, the average that suits growth rates, ratios and scale factors of any kind.

Harmonic Mean of Two Numbers

H = \frac{2ab}{a + b}

AlgebraThe reciprocal-based average of two numbers, the right average for rates such as speeds covering equal distances.

Inverse Variation

y = \frac{k}{x}

AlgebraRelation in which y falls as x rises so that their product stays at a constant k, the shape of Boyle's law and gear ratios.

Cramer's Rule for x (2×2 System)

x = \frac{c_1 b_2 - b_1 c_2}{a_1 b_2 - b_1 a_2}

AlgebraCramer's rule for the x that solves a pair of simultaneous linear equations, written as a ratio of two determinants.

Cramer's Rule for y (2×2 System)

y = \frac{a_1 c_2 - c_1 a_2}{a_1 b_2 - b_1 a_2}

AlgebraCramer's rule for the y that solves a pair of simultaneous linear equations, written as a ratio of two determinants.

Magnitude of a 2D Vector

|\vec{v}| = \sqrt{v_x^2 + v_y^2}

Vectors & MatricesAlgebraGeometryFinds the length of a two-dimensional vector from its x- and y-components, the Pythagorean theorem written for arrows instead of triangles.

Magnitude of a 3D Vector

|\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}

Vectors & MatricesAlgebraGeometryFinds the length of a three-dimensional vector from its x-, y-, and z-components, extending Pythagoras into space.

Direction Angle of a 2D Vector

\theta = \operatorname{atan2}(v_y,\; v_x)

Vectors & MatricesTrigonometryAlgebraGives the direction a two-dimensional vector points, measured counterclockwise from the positive x-axis, from its two components.

Unit Vector Component (Normalization)

\hat{u}_x = \frac{v_x}{|\vec{v}|}

Vectors & MatricesAlgebraScales one component of a vector down to the matching component of the unit vector that points the same way, by dividing by the magnitude.

Dot Product of Two 2D Vectors (Components)

\vec{a}\cdot\vec{b} = a_x b_x + a_y b_y

Vectors & MatricesAlgebraMultiplies two plane vectors component by component and adds the results, giving the scalar that measures how much they share a direction.

Dot Product of Two 3D Vectors (Components)

\vec{a}\cdot\vec{b} = a_x b_x + a_y b_y + a_z b_z

Vectors & MatricesAlgebraMultiplies two space vectors component by component and sums the three products, the standard test for perpendicularity in three dimensions.

Angle Between Two 2D Vectors (Components)

\theta = \arccos\left(\frac{a_x b_x + a_y b_y}{\sqrt{a_x^2+a_y^2}\,\sqrt{b_x^2+b_y^2}}\right)

Vectors & MatricesTrigonometryAlgebraFinds the angle separating two plane vectors directly from their four components, by way of the normalized dot product.

Cross Product z-Component of Two 2D Vectors

(\vec{a}\times\vec{b})_z = a_x b_y - a_y b_x

Vectors & MatricesAlgebraGeometryComputes the signed out-of-plane cross product of two plane vectors from their components, whose sign reveals their turning direction.

Scalar Projection of One Vector onto Another

\text{comp}_{\vec{b}}\vec{a} = \frac{\vec{a}\cdot\vec{b}}{|\vec{b}|}

Vectors & MatricesAlgebraPhysicsMeasures how far a vector reaches along the direction of another, the length of the shadow it casts on that second vector.

Vector Subtraction (Component Form)

c_x = a_x - b_x

Vectors & MatricesAlgebraPhysicsSubtracts one vector from another one component at a time, the operation behind every relative-velocity and displacement-change problem.

Magnitude of a Scaled Vector

|k\vec{v}| = |k|\,|\vec{v}|

Vectors & MatricesAlgebraGives the length of a vector after multiplying it by a scalar, which stretches or shrinks the arrow without ever bending it.

Determinant of a 2×2 Matrix

D = ad - bc

Vectors & MatricesAlgebraComputes the determinant of a two-by-two matrix as the difference of its diagonal products, the number that decides invertibility.

Determinant of a 3×3 Matrix

D = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})

Vectors & MatricesAlgebraComputes the determinant of a three-by-three matrix by cofactor expansion along the first row, using all nine entries.

Parallelogram Area from Two Vectors

A = \left|a_x b_y - a_y b_x\right|

Vectors & MatricesGeometryAlgebraFinds the area of the parallelogram spanned by two plane vectors directly from their four components, no angle or height required.

Triangle Area from Two Vectors

A = \tfrac{1}{2}\left|a_x b_y - a_y b_x\right|

Vectors & MatricesGeometryAlgebraFinds the area of the triangle formed by two plane vectors from a shared vertex, using only their components — the shoelace formula.

Scalar Triple Product (Parallelepiped Volume)

V = \left|\vec{a}\cdot(\vec{b}\times\vec{c})\right|

Vectors & MatricesGeometryAlgebraFinds the volume of the parallelepiped spanned by three space vectors, from the nine components, as the size of their scalar triple product.

Inverse of a 2×2 Matrix (Top-Left Element)

(M^{-1})_{11} = \frac{d}{ad - bc}

Vectors & MatricesAlgebraGives the top-left entry of the inverse of a two-by-two matrix: the opposite diagonal entry divided by the determinant.

Trace of a 2×2 Matrix

\operatorname{tr}(M) = a_{11} + a_{22}

Vectors & MatricesAlgebraAdds the two entries on the main diagonal of a square matrix, a quantity unchanged by any change of basis.

Z-Score (Standard Score)

z = \frac{x - \mu}{\sigma}

StatisticsAlgebraHow many standard deviations a value sits above or below the mean, turning any measurement into a comparable standard score.

Variance and Standard Deviation

\sigma^2 = \sigma \cdot \sigma

StatisticsAlgebraConverts between variance and standard deviation: the variance is the square of the standard deviation, and the deviation is its square root.

Coefficient of Variation

CV = \frac{s}{\bar{x}}

StatisticsAlgebraRelative variability: the standard deviation expressed as a fraction of the mean, so spreads measured on different scales can be compared.

Standard Error of the Mean

SE = \frac{\sigma}{\sqrt{n}}

StatisticsAlgebraHow much a sample mean typically wanders from the true mean, shrinking with the square root of the sample size.

Margin of Error for a Mean

E = z \frac{\sigma}{\sqrt{n}}

StatisticsAlgebraHalf-width of a confidence interval for a mean, built from the critical z-value, the standard deviation and the sample size.

Confidence Interval Lower Limit

L = \bar{x} - z \frac{\sigma}{\sqrt{n}}

StatisticsAlgebraThe lower bound of a confidence interval for a mean, pulling the critical z-value and standard error back from the sample mean.

Confidence Interval Upper Limit

U = \bar{x} + z \frac{\sigma}{\sqrt{n}}

StatisticsAlgebraThe upper bound of a confidence interval for a mean, adding the critical z-value times the standard error to the sample mean.

Sample Size for a Mean

n = \left( \frac{z \sigma}{E} \right)^{2}

StatisticsAlgebraHow many observations a study needs to estimate a mean within a target margin of error at a chosen confidence level.

Sample Size for a Proportion

n = \frac{z^{2} \, p (1 - p)}{E^{2}}

StatisticsAlgebraHow many respondents a survey needs to estimate a percentage within a target margin of error at a chosen confidence level.

Standard Error of a Proportion

SE = \sqrt{\frac{p (1 - p)}{n}}

StatisticsAlgebraHow much a sample percentage typically wanders from the true population proportion, given the proportion and the sample size.

Margin of Error for a Proportion

E = z \sqrt{\frac{p (1 - p)}{n}}

StatisticsAlgebraThe plus-or-minus quoted with a poll result, built from the critical z-value, the sample proportion and the number of respondents.

One-Sample Z-Test Statistic

z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}

StatisticsAlgebraTests a sample mean against a claimed population mean when the population standard deviation is known, in standard-error units.

One-Sample T-Test Statistic

t = \frac{\bar{x} - \mu}{s / \sqrt{n}}

StatisticsAlgebraTests a sample mean against a claimed value when the standard deviation is estimated from the sample itself rather than known.

Two-Sample Z-Test Statistic

z = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\dfrac{\sigma_1^{2}}{n_1} + \dfrac{\sigma_2^{2}}{n_2}}}

StatisticsAlgebraCompares two independent sample means when both population standard deviations are known, scaled by the combined standard error.

Pooled Standard Deviation

s_p = \sqrt{\frac{(n_1 - 1) s_1^{2} + (n_2 - 1) s_2^{2}}{n_1 + n_2 - 2}}

StatisticsAlgebraCombines two sample standard deviations into a single estimate of common spread, weighting each by its degrees of freedom.

Degrees of Freedom (One-Sample t)

df = n - 1

StatisticsAlgebraThe degrees of freedom used to look up a critical value for a one-sample t-test or confidence interval, one fewer than the sample size.

Cohen's d (Effect Size)

d = \frac{\bar{x}_1 - \bar{x}_2}{s_p}

StatisticsAlgebraStandardised effect size: the gap between two group means measured in pooled standard deviations rather than raw units.

Range (Max minus Min)

R = x_{\max} - x_{\min}

StatisticsAlgebraThe simplest measure of spread in a data set: the distance from the smallest observation to the largest.

Midrange

M = \frac{x_{\max} + x_{\min}}{2}

StatisticsAlgebraThe midpoint between the largest and smallest observations, a quick centre estimate computed from just the two extremes.

Interquartile Range (IQR)

IQR = Q_3 - Q_1

StatisticsAlgebraThe width of the middle half of a data set, from the first quartile to the third, and the spread measure box plots are built on.

Outlier Lower Fence

LF = Q_1 - 1.5 \, IQR

StatisticsAlgebraTukey's lower cutoff for outliers: any observation below one and a half interquartile ranges under the first quartile is flagged.

Outlier Upper Fence

UF = Q_3 + 1.5 \, IQR

StatisticsAlgebraTukey's upper cutoff for outliers: any observation above one and a half interquartile ranges beyond the third quartile is flagged.

Regression Slope from Correlation

b = r \frac{s_y}{s_x}

StatisticsAlgebraThe least-squares slope of a regression line, recovered from the correlation coefficient and the two standard deviations.

Regression Line Intercept

a = \bar{y} - b \bar{x}

StatisticsAlgebraThe y-intercept of a least-squares line, fixed by the requirement that the line pass through the point of averages.

Predicted Value from a Regression Line

\hat{y} = a + b x

StatisticsAlgebraReads a prediction off a fitted least-squares line for any chosen value of the predictor, given the intercept and slope.

Coefficient of Determination (R²)

R^{2} = r^{2}

StatisticsAlgebraThe share of variation in y explained by a simple linear regression, obtained by squaring the correlation coefficient.

Chi-Square Contribution of One Cell

\chi^{2}_{\text{cell}} = \frac{(O - E)^{2}}{E}

StatisticsAlgebraHow much a single cell of a contingency or goodness-of-fit table adds to the chi-square statistic, from its observed and expected counts.

Normal Probability Density

f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x - \mu)^{2}}{2\sigma^{2}}}

StatisticsAlgebraThe height of the normal bell curve at a given value, set by the distance from the mean in standard deviations.

Percent Error

PE = \frac{|x_{\text{meas}} - x_{\text{acc}}|}{x_{\text{acc}}}

StatisticsAlgebraHow far a measurement strays from the accepted value, expressed as a fraction of that accepted value for lab reports and calibration checks.

Percent Difference

PD = \frac{|x_1 - x_2|}{(x_1 + x_2)/2}

StatisticsAlgebraCompares two measurements of equal standing by dividing their gap by their average, when neither counts as the accepted value.

Weighted Mean of Two Groups

\bar{x} = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2}

StatisticsAlgebraCombines the averages of two groups into one overall mean, weighting each group by how many observations it contains.

Factorial

n! = n \times (n-1) \times \cdots \times 2 \times 1

ProbabilityAlgebraThe product of every whole number from n down to 1, counting the ways that n distinct objects can be arranged in order.

Permutations (nPr)

{}_{n}P_{r} = \frac{n!}{(n-r)!}

ProbabilityAlgebraNumber of ways to choose r items from n distinct items when the order of the selection matters, as with podium finishes.

Combinations (nCr)

{}_{n}C_{r} = \binom{n}{r} = \frac{n!}{r! \, (n-r)!}

ProbabilityAlgebraNumber of ways to choose r items from n distinct items when the order does not matter, as with poker hands or lottery tickets.

Arrangements with Repetition

N = n^{r}

ProbabilityAlgebraNumber of sequences of length r drawn from n available choices when repeats are allowed, as with PIN codes and passwords.