📊 Algebra Calculators

Free online algebra calculators and tools. Choose a calculator below.

All Algebra Tools

Quadratic Equation Solver Solve quadratic equations (ax² + bx + c = 0). Find real and complex roots using ... Linear Equation Solver Solve systems of two linear equations with two unknowns. Find x and y values ins... Exponent Calculator Calculate powers and exponents. Compute x raised to the power of n with any base... Logarithm Calculator Calculate logarithms with any base. Compute log, ln, log₂, and log₁₀ of any numb... Factorial Calculator Calculate the factorial of any number (n!). Also computes double factorial and s...

Solving Quadratic Equations

A quadratic equation has the form ax² + bx + c = 0 where a ≠ 0. The three standard solution methods are factoring, completing the square, and the quadratic formula. The quadratic formula x = (−b ± √(b² − 4ac)) ÷ 2a always works regardless of whether the equation factors neatly. The discriminant b² − 4ac determines the nature of the roots: if positive, two distinct real roots exist; if zero, one repeated real root (the parabola touches the x-axis at exactly one point); if negative, two complex conjugate roots (no real x-intercepts). Quadratic equations model projectile motion, profit maximisation, and any physical system where rate of change is proportional to the current value.

Exponents and Logarithms

Exponents and logarithms are inverse operations. If b^x = y, then log_b(y) = x. The natural logarithm ln uses base e ≈ 2.71828; log₁₀ uses base 10; log₂ (binary logarithm) is used in computing. Key exponent rules: x^a × x^b = x^(a+b), (x^a)^b = x^(ab), x^0 = 1 for any x ≠ 0, and x^(−n) = 1/x^n. Key logarithm rules: log(ab) = log(a) + log(b), log(a/b) = log(a) − log(b), log(a^n) = n·log(a). The change-of-base formula log_b(x) = ln(x) ÷ ln(b) lets you compute any logarithm using a calculator's natural log or log₁₀ keys.

Factorials and Combinatorics

The factorial n! = n × (n−1) × (n−2) × ... × 2 × 1 counts the number of ways to arrange n distinct objects. By convention, 0! = 1. Factorials grow extremely rapidly: 10! = 3,628,800 and 20! ≈ 2.4 × 10¹⁸. Permutations count ordered arrangements: P(n,r) = n! ÷ (n−r)! gives the number of ways to choose r items from n and arrange them in order. Combinations count unordered selections: C(n,r) = n! ÷ (r! × (n−r)!) gives the number of ways to choose r items from n without regard to order. These are foundational to probability theory — the probability of an event equals the number of favourable outcomes divided by the total number of equally likely outcomes.

Linear Equations and Systems

A linear equation in two variables (ax + by = c) represents a straight line on the coordinate plane. A system of two linear equations has one of three outcomes: one unique solution (lines intersect at a point), no solution (lines are parallel and never meet), or infinitely many solutions (lines are identical). The substitution method works by solving one equation for one variable and substituting into the other. The elimination method adds or subtracts multiples of the equations to cancel one variable. For systems of three or more equations, matrix methods (Gaussian elimination, Cramer's rule) are more systematic. Linear systems arise throughout engineering, economics, and data science — any time multiple constraints must be satisfied simultaneously.