📐 Algebraic Polynomial & Parabola Solver
Quadratic Equation Solver
Solve $ax^2 + bx + c = 0$ with step-by-step discriminant evaluation, real/complex roots, and parabola vertex analysis.
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Sample Equations:
x² − 5x + 6 = 0 (Roots: 2, 3)
x² − 4x + 4 = 0 (Repeated: 2)
x² + 4 = 0 (Complex: ±2i)
2x² + 4x − 6 = 0 (Roots: 1, −3)
Solutions for x
x₁ = 3, x₂ = 2
Parabola Geometric Properties:
Discriminant (Δ)
1
Vertex (h, k)
(2.5, −0.25)
Axis of Symmetry
x = 2.5
Parabola Opening
Upward (Minimum)
The Standard Quadratic Formula
For any second-degree polynomial equation $ax^2 + bx + c = 0$ (where $a \neq 0$), solutions are given by the quadratic formula:
x = [ −b ± √(b² − 4ac) ] / (2a)
The term under the radical $\Delta = b^2 - 4ac$ is the discriminant. It dictates whether the parabola crosses the x-axis twice ($\Delta > 0$), touches it at a single tangent vertex ($\Delta = 0$), or does not intersect the real plane ($\Delta < 0$).