Quadratic Equation & Parabola Geometry Calculator
Solve real & complex roots, extract analytical parabolic conic geometry, and reconstruct polynomials from key features
Polynomial Coefficients: a·x² + b·x + c = 0
Curvature (a ≠ 0)
Linear term
y-intercept
Equation Roots (x₁, x₂)
x₁ = 3.000, x₂ = 2.000
Two Distinct Real Roots (Δ > 0)
Factored Form: (x – 3.000)(x – 2.000) = 0
Discriminant (Δ = b² – 4ac)
1.000
(-5)² – 4(1)(6) = 25 – 24
Sum of Roots (Vieta: -b/a)
5.000
x₁ + x₂ = -(-5.0) / 1.0
Product of Roots (Vieta: c/a)
6.000
x₁ · x₂ = 6.0 / 1.0
Completing the Square Form
(x – 2.500)² = 0.250
(x + b/(2a))² = (b² – 4ac)/(4a²)
Parabolic Vertex & Extrema Point (h, k)
(2.500, -0.250)
Absolute Global Minimum
Axis of Symmetry: x = 2.500 • Opens Upward
Focus Coordinates (h, k + 1/(4a))
(2.500, 0.000)
Focal distance p = 0.250
Directrix Line (y = k – 1/(4a))
y = -0.500
Equidistant guide line
Latus Rectum Length (|1/a|)
1.000
Chord width through focus (4p)
Vertex Standard Form
y = (x – 2.500)² – 0.250
y = a(x – h)² + k
y-Intercept Point
(0, 6.000)
Value evaluated at x = 0
Semi-Latus Rectum (2p = |1/(2a)|)
0.500
Half-width focal parameter
Point Evaluation & Tangent Line Analysis
Evaluate value & tangent line slope
Reconstructed Standard Equation
2x² – 5x – 12 = 0
a = 2.000, b = -5.000, c = -12.000
Function Value at x₀: f(x₀)
-14.000
Coordinate: (2.000, -14.000)
Instantaneous Tangent Slope f'(x₀)
3.000
Derivative: 2ax₀ + b
Tangent Line Equation
y = 3x – 20
Linear Taylor polynomial at x₀
Normal Line Equation (Perpendicular)
y = -0.333x – 13.333
Slope: m_norm = -1 / f'(x₀)