Math and measurement tools
Quadratic Formula Calculator
Solve ax² + bx + c = 0 with real or complex roots. See the discriminant, vertex, symmetry axis and method, including linear equations when a is zero.
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How to use this tool
1. Rearrange the equation as ax² + bx + c = 0. Enter the signed coefficients, including 1 or -1 for an implied coefficient of x².
2. Select Solve equation to find approximate roots, classify the discriminant and calculate the vertex and symmetry axis.
3. Inspect the formula and explanation. If a is zero, the tool handles the resulting linear or constant equation explicitly. Copy or print your coefficients and results.
The quadratic formula
For nonzero a, x = (−b ± √(b² − 4ac)) ÷ (2a). The discriminant b² − 4ac determines the root type: positive gives two distinct real roots, zero gives one repeated real root, and negative gives two complex conjugate roots.
Worked example
For x² − 3x + 2 = 0, enter a = 1, b = -3 and c = 2. The discriminant is 9 − 8 = 1, giving roots 1 and 2. The vertex is (1.5, -0.25), the symmetry axis is x = 1.5 and the parabola opens upward.
Questions and assumptions
Complex roots
A negative discriminant does not mean that the equation has no solution over the complex numbers. The tool displays a conjugate pair in the form real part ± imaginary magnitude × i, where i² = −1. For x² + 1 = 0, the roots are 0 ± 1i.
What happens when a is zero?
The equation becomes bx + c = 0. If b is nonzero, x = −c ÷ b. If both a and b are zero, c = 0 is true for every real x; a nonzero c makes the equation inconsistent. These cases have no parabola vertex.
Precision and numerical stability
Use coefficients from -1 billion to 1 billion with up to six decimal places. The discriminant sign is calculated using exact fixed-point integers. Real roots use a numerically stable rearrangement of the formula to reduce cancellation. Decimal roots and vertex values are rounded to 12 significant digits; symbolic radical simplification, fractions and graph plotting are outside this tool’s scope.
Sources and further reading
Built and maintained by Custom Logic. Page updated .