Algebra Calculators
Free algebra calculators: quadratic formula, quadratic inequalities, linear equation solver, polynomial calculator, absolute value, and systems of equations solver.
Algebra Calculators - Equations and Systems
Algebra is the branch of mathematics that uses symbols (variables) to represent numbers in equations and formulas. From single-variable equations to systems of equations, algebra is the language of quantitative reasoning. These calculators solve algebraic equations step-by-step, making the method as clear as the answer.
Graphing Quadratic Inequalities Calculator - Solve any quadratic inequality ax² + bx + c > 0, < 0, ≥ 0, or ≤ 0. Enter coefficients a, b, c and choose the inequality direction. Outputs discriminant, boundary roots, solution set in plain language, and interval notation. Handles all three discriminant cases: two real roots, repeated root, and no real roots. - Solve any quadratic equation ax² + bx + c = 0 using the quadratic formula. Displays both roots (real or complex), discriminant with interpretation, vertex, axis of symmetry, and full step-by-step working.
Linear Equation Solver - Solve single-variable equations in the form ax + b = c, and two-variable systems of linear equations (2×2) using Cramer’s rule. Shows complete step-by-step working including the elimination process and verification. Handles no-solution and infinite-solution cases with explanations.
Polynomial Calculator - Solve quadratic equations (ax² + bx + c = 0) with exact roots and discriminant, find real roots of cubic equations numerically, and evaluate any polynomial P(x) at a given x value. Shows full step-by-step working for all modes.
Absolute Value Calculator - Find |x| for any number, evaluate expressions containing absolute value bars such as |3x − 2|, and solve absolute value inequalities |ax + b| < c or |ax + b| > c. Results shown in interval notation with complete step-by-step working.
Systems of Equations Solver - Solve 2×2 linear systems (two equations, two unknowns) using Cramer’s Rule with full determinant working, and 3×3 systems using Gaussian elimination. Detects parallel-lines (no solution) and dependent-equations (infinite solutions) cases automatically.
Matrix Calculator - Perform matrix addition, subtraction, multiplication, determinant, inverse, and transpose for 2×2 and 3×3 matrices. Shows step-by-step cofactor expansion for determinants and detects singular matrices (det = 0) automatically.
Why Step-by-Step Matters
Algebra calculators are most valuable when they show the method, not just the answer. Each solved equation is an opportunity to reinforce the underlying technique: isolating the variable using inverse operations, applying Cramer’s rule for systems, and verifying solutions by substitution. Use these calculators to check your work or to understand the method when you are stuck.
Frequently Asked Questions
What is a linear equation?
A linear equation is an equation where the variable appears only to the first power (no x², √x, etc.). It graphs as a straight line. One-variable form: ax + b = c. Two-variable form: ax + by = c (a line in the coordinate plane). Solving a two-variable system means finding where two lines intersect.
What is Cramer's Rule?
Cramer's Rule solves a 2×2 system a₁x + b₁y = c₁ and a₂x + b₂y = c₂ using the determinant D = a₁b₂ − a₂b₁. Then x = (c₁b₂ − c₂b₁) / D and y = (a₁c₂ − a₂c₁) / D. If D = 0, the system has no unique solution (either no solution or infinitely many).
What should I do if the equation has no solution?
A one-variable equation has no solution when a = 0 but b ≠ c (e.g., 0x + 3 = 5 → 3 = 5, which is false). A two-variable system has no solution when the two lines are parallel (same slope, different y-intercepts). In both cases, the calculator identifies and explains the outcome.