Normal-approximation density split at the real-root boundaryLive current inputs
Math & Statistics
Quadratic Equation Distribution Calculator
Propagate independent uncertainty in coefficients a, b, and c into the discriminant of ax^2 + bx + c = 0, estimate the normal-approximation chance of real roots, and show which coefficient contributes most to discriminant variance.
Mean-coefficient discriminant-
Propagated discriminant uncertainty-
Approximate chance discriminant is nonnegative-
Mean-coefficient root class-
Largest variance contributor-
Displayed discriminant interval-
Mean function value at probe x-
Approximation boundary-
DISCRIMINANT UNCERTAINTY DISTRIBUTION
Normal-approximation density split at the real-root boundary
The density is shaded on both sides of Delta = 0, while a contribution strip shows how uncertainty in a, b, and c enters the propagated variance.
COEFFICIENT SENSITIVITY REGISTER
One-standard-uncertainty coefficient scenarios
Each row perturbs one coefficient while holding the others at their means, revealing the local discriminant response.
| Scenario | a | b | c | Discriminant | Root class | Change from mean |
|---|
DISTRIBUTION SETUP
Enter coefficient uncertainty on a consistent scale
- Keep a nonzero and ensure its uncertainty is small enough that sign changes are unlikely.
- Use standard uncertainties, not full ranges.
- Enter uncertainties on the same coefficient scale as a, b, and c.
- Use the density as a local approximation around the mean coefficients.
- Inspect nonlinear or correlated cases with simulation or exact analysis.
THE BOUNDARY IS DELTA ZERO
Root type changes abruptly even when coefficients change smoothly
The discriminant is continuous in the coefficients, but the real-versus-complex classification changes at zero. The density makes proximity to that boundary visible.
First-order propagation can be inaccurate with large uncertainty, strong nonlinearity, coefficient correlation, or appreciable probability that a approaches zero.
UNCERTAINTY PROPAGATION
Apply the delta method to the quadratic discriminant
For independent coefficient uncertainties, the gradient of Delta = b^2 - 4ac provides a first-order variance approximation. A normal CDF then estimates the share of the approximating distribution at or above zero.
Detailed calculation process and general formulas
Delta = b^2 - 4acdDelta/da = -4c; dDelta/db = 2b; dDelta/dc = -4aVar(Delta) ~= (-4c)^2 s_a^2 + (2b)^2 s_b^2 + (-4a)^2 s_c^2sigma_Delta = sqrt(Var(Delta))P(real roots) ~= Phi(Delta / sigma_Delta)Symbols, meanings, and units
- a,b,c
- mean quadratic coefficientscoefficient units
- Delta
- quadratic discriminantsquared coefficient scale
- s_a,s_b,s_c
- independent standard coefficient uncertaintiescoefficient units
- sigma_Delta
- first-order discriminant standard uncertaintydiscriminant units
- Phi
- standard normal cumulative distribution functiondimensionless
UNCERTAINTY READING
Three outputs explain the approximation
The page reports location, spread, and sensitivity.
Discriminant center
-Mean coefficients determine the nominal root class.
Propagated spread
-The gradient maps coefficient uncertainty into discriminant uncertainty.
Dominant contributor
-Variance shares show which coefficient uncertainty matters most locally.
Decision takeaway: When the density crosses zero substantially, report root-type uncertainty instead of only the nominal roots.
APPROXIMATION CHECKS
Conditions that require a stronger method
- Coefficient correlation
- Large relative uncertainty
- Possible sign change of coefficient a
- Non-normal coefficient distributions
- Rounded or interval-only coefficients
- Need for exact root distributions
Applied decisions
Two ways coefficient uncertainty changes root interpretation
Discriminant well above zero
The mean discriminant is large relative to propagated uncertainty.
What the result clarifies: Most of the approximating density remains in the two-real-root region.
Mean near the boundary
Small coefficient uncertainty places material density on both sides of zero.
What the result clarifies: A single nominal root classification hides meaningful uncertainty.
Worked default scenario
Current-input substitution and reconciliation
Method references
Evidence used to frame this specific model
Scope and limitations
This calculator uses first-order independent-error propagation and a normal approximation. It is not exact when coefficient uncertainties are large, correlated, non-normal, or allow a to approach zero. Use simulation, interval analysis, or symbolic methods when those conditions matter.
Quadratic Equation Distribution Calculator | Discriminant Uncertainty FAQ
Why approximate the discriminant instead of the roots directly?
Root type is determined by the discriminant, and its gradient provides a clear first-order uncertainty calculation.
What happens when sigma Delta is zero?
The coefficients are treated as fixed and the root class follows the nominal discriminant.
Are the root probabilities exact?
No. They come from a local normal approximation to the discriminant.
What if coefficient errors are correlated?
Covariance terms are required; this independent-input model is not sufficient.