Algebra & Equations

Find the midpoint between two points (x₁, y₁) and (x₂, y₂) on the coordinate plane. Free midpoint formula calculator showing both x and y coordinates.

Calculate percent error between a measured value and the actual (true) value. Free percentage error calculator for science labs, chemistry, and physics.

Solve right triangles with the Pythagorean theorem. Find the hypotenuse from two legs, or a missing leg from the hypotenuse and one leg.

Solve ax² + bx + c = 0 with the quadratic formula — real and complex roots, plus the discriminant.

Find the slope of a line between two points, plus the angle of inclination and y-intercept. Free slope and rise-over-run calculator with the line equation.

Find the eigenvalues of a 2x2 matrix from its characteristic equation. Enter the four entries to get real or complex roots.

Solve |ax+b|=c for x and graph y=|ax+b| against y=c to see the intersection points.

Find the angle between two vectors using the dot product formula. Enter components to get the angle in degrees and radians.

Find the nth term and the sum of an arithmetic sequence from the first term and common difference.

Expand (a+b)^n with the binomial theorem and list every term coefficient.

Find the cofactor matrix and adjugate (adjoint) of a 2x2 or 3x3 matrix. Enter rows of numbers to get both matrices.

Rewrite ax^2+bx+c in vertex form a(x-h)^2+k and read off the vertex.

Solve a 2x2 or 3x3 system of linear equations with Cramer’s rule using determinants. Enter coefficients and constants.

Compute the cross product of two 3D vectors. Enter three components each to get the perpendicular vector and its magnitude.

Solve ax^3+bx^2+cx+d=0 for all real and complex roots, with the cubic curve graphed.

Find the constant of variation k and predict a new value using direct or inverse proportionality.

Find the discriminant b^2-4ac of a quadratic and instantly see how many real roots it has.

Calculate the straight-line distance between two points (x₁, y₁) and (x₂, y₂) using the distance formula derived from the Pythagorean theorem.

Calculate the dot (scalar) product of two vectors of equal length. Enter components separated by spaces or commas.

Factor ax^2+bx+c into a(x-r1)(x-r2) using its roots, with real or complex output.

Compose two linear functions f and g and evaluate (f∘g)(x) at a point.

Compute the nth term, finite sum, and infinite sum of a geometric sequence from its ratio.

Apply exponent laws to same-base powers: multiply, divide, or raise to a power and simplify.

Find the equation of a line from two points — slope-intercept (y = mx + b), point-slope and standard form, plus the slope and intercepts, with a graph.

Compute a linear combination c1*v1 + c2*v2 + ... of several vectors. Enter vectors as rows and matching scalar coefficients.

Solve any linear equation of the form ax + b = 0 for x instantly. Free step-by-step linear equation calculator with edge-case handling.

Solve a one-variable linear inequality ax+b (</<=/>/>=) c and view the solution region on a number line.

Add two matrices of the same size element by element. Enter each matrix as rows of space-separated numbers and get the resulting matrix instantly.

Compute the determinant of a 2x2 or 3x3 matrix entered as rows of numbers; auto-detects the size.

Find the inverse of a 2x2 or 3x3 matrix using the determinant and adjugate. Enter rows of numbers; must be non-singular.

Multiply two matrices using the row-by-column dot product rule. Enter both matrices as rows of numbers; sizes must be compatible.

Raise a square matrix to an integer power. Enter the matrix as rows of numbers and an exponent to compute A to the n.

Find the rank of any matrix using Gaussian elimination. Enter rows of numbers to count the independent rows (nonzero pivots).

Subtract one matrix from another element by element. Enter both matrices as rows of numbers and see the difference matrix with each step.

Calculate the trace of a square matrix: the sum of its main diagonal entries. Enter the matrix as rows of numbers.

Find the transpose of any matrix by flipping rows and columns. Enter rows of space-separated numbers and get A-transpose instantly.

Find the quadratic equation y = ax² + bx + c of the parabola through three points, plus its vertex, axis of symmetry and discriminant, with a graph.

Find the equations of the lines parallel and perpendicular to a given slope that pass through a point, in slope-intercept form, with a graph.

Decompose (px+q)/((x-r1)(x-r2)) into A/(x-r1)+B/(x-r2) for distinct linear factors.

Write the equation of a line from a point and a slope using point-slope form y − y₁ = m(x − x₁), then convert to slope-intercept and standard form, with a graph.

Evaluate any polynomial P(x) at a given x value using Horner's method, with the curve plotted around the point.

Solve a proportion a/b = c/x for the unknown x using cross multiplication. Free proportion and ratio solver for equivalent fractions.

List every possible rational root of a polynomial from its constant and leading coefficients.

Reduce any matrix to reduced row echelon form with Gauss-Jordan elimination. Enter rows of numbers to see the RREF.

Multiply a matrix by a scalar. Enter the matrix as rows of numbers and a scalar value to scale every entry instantly.

Compute the scalar triple product of three 3D vectors and the volume of the parallelepiped they form.

Divide a polynomial by (x-c) using synthetic division to get the quotient and remainder.

Solve a 2x2 system of linear equations using Cramer’s rule. Enter coefficients to find x and y instantly, with determinant and edge-case detection.

Normalize a vector to length 1. Enter the components and get the unit vector pointing in the same direction plus its magnitude.

Find the magnitude (length/norm) of a vector in any dimension. Enter components separated by spaces or commas to get |v|.

Project one vector onto another. Enter vectors a and b to get the projection vector and the scalar projection of a onto b.