Quadratic Equation Solver
Solve any quadratic equation of the form ax² + bx + c = 0. Enter the coefficients below to find the roots (real or complex).
Your results will appear here.
Formula
For a quadratic equation ax² + bx + c = 0 (where a ≠ 0):
Quadratic Formula:
x = (−b ± √(b² − 4ac)) / (2a)
Discriminant: Δ = b² − 4ac
- Δ > 0 → Two distinct real roots
- Δ = 0 → One repeated real root (double root)
- Δ < 0 → Two complex conjugate roots
Vieta's Formulas:
- Sum of roots: x₁ + x₂ = −b / a
- Product of roots: x₁ · x₂ = c / a
Vertex of the parabola: x = −b / (2a)
Assumptions & References
- Coefficient a must be non-zero; if a = 0 the equation is linear, not quadratic.
- All coefficients (a, b, c) must be real numbers.
- When Δ < 0, roots are complex numbers of the form α ± βi and cannot be plotted on the real number line.
- Results are rounded to 6 decimal places for display; full precision is used internally.
- Formula source: Algebra and Trigonometry, Stewart, Redlin & Watson; standard quadratic formula derivation via completing the square.
- Complex root representation follows the standard convention: i = √(−1).