Quadratic Formula Calculator
Last updated: November 4, 2026
Solve ax² + bx + c = 0, classify real or complex roots, and handle linear degeneracies.
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Quick summary: Solve ax² + bx + c = 0, classify real or complex roots, and handle linear degeneracies.
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Updated reference sections in this guide are in English.
Using this calculator
Write your equation with zero on the right before entering the coefficients. Include signs: x² − 3x + 2 = 0 uses a = 1, b = −3, c = 2. The result distinguishes two real roots, a repeated root and a complex conjugate pair. It also handles a = 0 explicitly.
Calculation method
For a ≠ 0: x = (−b ± √D) / (2a)
For D < 0: x = −b/(2a) ± i√(−D)/(2|a|)
The real-root implementation uses a numerically stable equivalent: compute one root through a sign-aware intermediate q, then obtain the other from c/q. Coefficients are scaled before the discriminant calculation to reduce avoidable overflow. These steps improve numeric evaluation without changing the equation.
Worked examples
Two real roots
For a = 1, b = −3, c = 2, D = 9 − 8 = 1. The roots are 1 and 2. Substitution gives zero for either value.
Complex and degenerate equations
x² + 1 = 0 gives 0 ± 1i. If a = 0, b = 2 and c = −8, the remaining equation 2x − 8 = 0 gives x = 4.
Check this scenario in the calculator:
- Coefficient a
- 1
- Coefficient b
- -3
- Coefficient c
- 2
Understanding the result
The ± symbol denotes two conjugate solutions. A discriminant of zero gives one repeated real root. When a and b are both zero, c = 0 makes the equation an identity, while a nonzero c makes it inconsistent. Those cases are reported as classifications rather than fabricated roots.
Boundaries and assumptions
Coefficients are numeric values, not algebraic expressions, and are limited to ±10¹². Browser floating-point arithmetic can affect a discriminant very close to zero or roots with extreme scales. Displayed roots are approximations; the tool does not preserve exact radicals or implement arbitrary-precision complex algebra.
Questions and reference
Why can a quadratic have complex roots?
A negative discriminant requires the square root of a negative quantity, represented using i² = −1.
Does a = 0 mean an error?
It means the equation is no longer quadratic. The remaining linear or constant equation is solved or classified.
Reference: OpenStax: quadratic equations.