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Quadratic Equation Calculator
Enter coefficients a, b, and c for ax² + bx + c = 0. The tool returns the discriminant, roots (real or complex), and a short nature summary—using the same high-precision habits described in our precision guide.
Solve ax² + bx + c = 0
Results
| Discriminant (b² − 4ac) | — |
|---|---|
| Root x₁ | — |
| Root x₂ | — |
| Nature of roots | — |
How to solve a quadratic equation
A quadratic equation has the form ax² + bx + c = 0 with a ≠ 0. The quadratic formula gives the roots directly:
- Compute the discriminant Δ = b² − 4ac.
- If Δ > 0, there are two distinct real roots: x = (−b ± √Δ) / (2a).
- If Δ = 0, there is one repeated real root: x = −b / (2a).
- If Δ < 0, roots are complex conjugates; this calculator displays them in a ± bi form.
When a equals zero
If a = 0, the equation is linear: bx + c = 0. When b ≠ 0, the single root is x = −c / b. The discriminant row is not used in that case.
Units, rounding, and limits
Coefficients may be any real numbers typed as text. Real roots are shown with up to 14 decimal places; trailing zeros are trimmed. Complex roots show a real part and an imaginary part in standard a ± bi notation. This is a school-algebra model—not a substitute for symbolic computer algebra in advanced research.
FAQ
What is the quadratic formula?
For ax² + bx + c = 0 with a ≠ 0, the roots are x = (−b ± √(b² − 4ac)) / (2a). The expression b² − 4ac is the discriminant; it tells you whether roots are real and distinct, real and repeated, or complex.
What happens when a equals zero?
When a = 0, the equation is linear: bx + c = 0. If b ≠ 0, there is one root x = −c / b. If b = 0 and c = 0, every real number satisfies the equation; if b = 0 and c ≠ 0, there is no solution.