Quadratic Equation Solver
Solve a*x^2 + b*x + c = 0 with explicit discriminant branches; degenerate a = 0 equations fall back to honest linear, no-solution or all-real-numbers cases.
Description
Solve a*x^2 + b*x + c = 0 with explicit discriminant branches; degenerate a = 0 equations fall back to honest linear, no-solution or all-real-numbers cases.
Solve a*x^2 + b*x + c = 0 with explicit discriminant branches; degenerate a = 0 equations fall back to honest linear, no-solution or all-real-numbers cases.
When to use Quadratic Equation Solver
Use this calculator for direct numerical work with the stated polynomial or equation form and for checking a hand-worked solution.
- a
- Coefficient of x^2; zero degrades to a linear equation.
- b
- Coefficient of x.
- c
- Constant term.
How the calculation works
The formula below determines the tool output. Case: Resolution case: two_real_roots, double_root, complex_roots, linear, no_solution or all_real_numbers. Discriminant: Discriminant b*b - 4*a*c; reported as 0 when a = 0. Roots: Real solutions sorted ascending; empty for complex or degenerate no/all-solution cases. Real part: Real part of the complex pair when case is complex_roots, else 0. Imaginary part: Nonnegative imaginary magnitude of the complex pair when case is complex_roots, else 0.1
roots = (−b ± √(b² − 4ac)) / (2a) Limitations and assumptions
Coefficients are evaluated with floating-point arithmetic. Ill-conditioned systems, nearly repeated roots, and coefficients with very different magnitudes can magnify representation error; the result is not a symbolic proof.
Alternative or Complementary approaches
Complement the numerical result by substituting it back into the original expression. Symbolic algebra or higher-precision arithmetic is a better alternative when an exact form or a proof is required.
References
-
Quadratic formula — Wikipedia contributors
Similar or alternative tools
- Polynomial Evaluator
Evaluate a polynomial at a point using Horner's method with highest-degree-first coefficients.
- Linear Equation System Solver
Solve a system of two linear equations in two unknowns with Cramer's rule.
- Synthetic Division Calculator
Divide a polynomial (highest-degree-first coefficients) by (x - r) using synthetic division, returning the quotient and remainder.