Equation Solver

Mastering Equation Solving: Complete Guide

Equations are the foundation of algebra and appear in every STEM field. This Professional Equation Solver handles linear, quadratic, cubic, and 2×2 systems of equations, providing not just the answer but a clear, step‑by‑step breakdown and a visual graph where applicable.

Linear Equations (ax + b = 0)

A linear equation has degree 1 and exactly one solution (if a ≠ 0). Isolate x by moving the constant term and dividing by the coefficient: x = -b/a. The graph is a straight line crossing the x‑axis at the solution.

Quadratic Equations (ax² + bx + c = 0)

Solved via the quadratic formula: x = [-b ± √(b² - 4ac)] / (2a). The discriminant Δ = b² - 4ac reveals the nature of roots: Δ > 0 → two real roots; Δ = 0 → one repeated root; Δ

Systems of Linear Equations (2×2)

A system of two equations with two unknowns can be solved using Cramer's rule. Compute the determinant D = a₁b₂ - a₂b₁. If D ≠ 0, a unique solution exists. If D = 0, the lines are parallel (no solution or infinite solutions). The calculator graphs both lines and highlights the intersection point.

Cubic Equations (ax³ + bx² + cx + d = 0)

Every cubic has at least one real root. The calculator finds all three roots (real and complex) numerically using Cardano's method. The graph helps visualize the real roots as x‑intercepts.