Equation Solver — Linear, Quadratic & System of Equations

Solve linear, quadratic, cubic and systems of equations step-by-step with full working.

About this calculator

An equation solver finds the roots (values of the variable that make the equation true) for linear, quadratic, cubic, and quartic polynomial equations. The solver also handles systems of linear equations using Gaussian elimination.

Polynomial equations are the foundation of almost every applied math problem — from physics motion equations to chemistry equilibria to engineering design. Hand-solving anything beyond a quadratic is tedious and error-prone; a computational solver is the standard tool.

Formula

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

  • Quadratic — ax² + bx + c = 0
  • Discriminant — b² − 4ac; positive: 2 real roots; zero: 1 repeated; negative: 2 complex
  • Cubic/Quartic — Use Cardano's or Ferrari's formula, or numerical methods

Worked example

Inputs: Solve x² − 5x + 6 = 0

  1. a = 1, b = −5, c = 6
  2. Discriminant = 25 − 24 = 1
  3. x = (5 ± 1) ÷ 2
  4. x = 3 or x = 2

Frequently asked questions

When are roots complex?

A polynomial equation has complex roots whenever the discriminant is negative. Complex roots always come in conjugate pairs for polynomials with real coefficients. Quadratics either have 0 or 2 complex roots; cubics always have at least 1 real root; quartics can have 0, 2, or 4 complex roots.

What about transcendental equations?

Equations involving sin, cos, exp, or log generally cannot be solved algebraically — they require numerical methods like Newton-Raphson iteration or bisection. Our equation solver covers polynomial cases; for transcendental equations use a graphing approach to find approximate roots first.

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