Quadratic Equation Solver

ax² + bx + c = 0 → roots via discriminant. Real or complex.

Inputs

Result

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How to use this calculator

  • Enter a, b, c.
  • Read discriminant + roots + vertex.

About this calculator

The quadratic formula x = (−b ± √(b² − 4ac)) / (2a) gives all roots of ax² + bx + c = 0. The discriminant Δ = b² − 4ac classifies: Δ > 0 → two real roots; Δ = 0 → one repeated root; Δ < 0 → two complex conjugate roots. The vertex x = −b/(2a) is the parabola's extreme point — minimum if a > 0, maximum if a < 0. Quadratics show up in projectile motion, optimization, and geometry whenever an x² term appears.

Frequently asked

What does discriminant tell me?+
The number and type of roots. >0 = two reals; =0 = one repeated; <0 = two complex conjugates.
When does a = 0?+
It's no longer quadratic — becomes linear (bx + c = 0). Calculator handles this gracefully.
Vertex use?+
Optimization. A profit function P(x) = −2x² + 100x − 800 maximizes at vertex x = 25.
Complex roots in real-world problems?+
Indicate no real solution (e.g. trajectory never reaches that height). In electrical engineering: oscillation modes.
Why factor when you can use formula?+
Formula always works. Factoring is faster when integer roots exist (e.g. x² − 5x + 6 = (x−2)(x−3) has roots 2, 3 by inspection).

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