Quadratic Equation Solver
ax² + bx + c = 0 → roots via discriminant. Real or complex.
Result
Two real roots
x = 2.000000, 1.000000
Discriminant = 1.0000 > 0.
- a1
- b-3
- c2
- Discriminantb² − 4ac = 1.0000
- x₁2.00000000
- x₂1.00000000
- Vertex(1.5000, -0.2500)
- Opensupward
Step-by-step
- Discriminant = b² − 4ac = -3² − 4×1×2 = 1.0000.
- Two distinct real roots via quadratic formula.
- Vertex: x = −b/(2a) = 1.5000; y = -0.2500.
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
The number and type of roots. >0 = two reals; =0 = one repeated; <0 = two complex conjugates.
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