Quadratic Equation Solver
Solve quadratic equations ax² + bx + c = 0 to find real or complex roots, the discriminant, and parabola vertex coordinates.
Run tool
This online quadratic solver calculates the solutions for any second-degree polynomial equation in the standard form ax² + bx + c = 0. Enter numeric values for the leading coefficient a and the constants b and c to determine the exact roots and geometric properties of the resulting parabola. When omitted, coefficient a defaults to 1, while b and c default to 0.
The calculator evaluates the discriminant to determine root behavior. A positive discriminant yields two distinct real solutions, a zero discriminant indicates a single repeated real root, and a negative discriminant calculates complex conjugate solutions with separate real and imaginary parts. In addition to roots, the calculation provides the coordinates of the parabola vertex (h, k). All computed outputs, including roots and vertex points, are rounded to ten decimal places for precision. Note that the leading coefficient a cannot equal zero, as quadratic equations require a non-zero squared term.
Instructions
- Enter the leading coefficient for A in the first field.
- Enter the linear coefficient for B in the second field, or leave it blank to use the default value 0.
- Enter the constant term for C in the third field, or leave it blank to use the default value 0.
- Click the calculate button to evaluate the roots, discriminant, and vertex coordinates.
Examples
- Solve x² - 5x + 6 = 0: enter A = 1, B = -5, and C = 6 to find two real roots x₁ = 3 and x₂ = 2 with discriminant 1 and vertex at (2.5, -0.25).
- Solve 2x² + 4x + 10 = 0: enter A = 2, B = 4, and C = 10 to find complex roots with real component -1 and imaginary component ±2, discriminant -64, and vertex at (-1, 8).
FAQ
What happens if coefficient A is set to 0?
Quadratic equations require a non-zero x² term. If coefficient A is set to 0, the equation becomes linear and the solver will reject the calculation.
How does the tool handle negative discriminants?
When the discriminant b² - 4ac is negative, the equation has no real solutions. The tool identifies the roots as complex and calculates the real and imaginary parts separately.
What precision is used for computed solutions?
All calculated values, including roots, discriminants, and vertex coordinates, are rounded to 10 decimal places.