Quadratic Equation Solver mechanism

Model scope

The stated root and vertex formulas are mathematically standard under a != 0. Review is required because the input contract and factoring behavior are incomplete; the instrument should explicitly state whether a = 0 is rejected or reduced to a linear equation, and whether factoring is over the integers, rationals, or reals.

• The quadratic-root formula is correct only when a != 0; the specification does not state this required constraint or how a = 0 is handled.
• The vertex formula is correct: h = -b/(2a), k = f(h), but it also requires a != 0.
• Factoring 'when rational' is underspecified: the coefficient domain, factor domain, and behavior when no rational factorization exists are not defined.
• Complex-aware output needs an explicit convention for sqrt of a negative discriminant and for formatting conjugate complex roots.

Units and conventions

• a, b, and c are real numeric coefficients and a != 0 for a quadratic input.
• If x carries units U and f(x) carries units V, then a has units V/U^2, b has units V/U, and c has units V; b^2 and 4ac therefore have matching units.
• Discriminant sign is meaningful only after coefficients are expressed in mutually consistent units.
• The test vectors use dimensionless coefficients and x values.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "a": 1,
    "b": -5,
    "c": 6
  },
  "expected_outputs": {
    "roots": [
      "2",
      "3"
    ],
    "vertex": [
      "5/2",
      "-1/4"
    ],
    "axis_of_symmetry": "x = 5/2",
    "discriminant": 1,
    "discriminant_sign": "positive",
    "factoring": "(x - 2)(x - 3)"
  },
  "working": "D = b^2 - 4ac = (-5)^2 - 4(1)(6) = 25 - 24 = 1. sqrt(D) = 1. x = (-b +/- sqrt(D))/(2a) = (5 +/- 1)/2, so x = 6/2 = 3 and x = 4/2 = 2. Vertex x-coordinate h = -b/(2a) = -(-5)/(2(1)) = 5/2. Vertex y-coordinate f(h) = (5/2)^2 - 5(5/2) + 6 = 25/4 - 25/2 + 24/4 = -1/4. Axis: x = 5/2. The polynomial factors as (x - 2)(x - 3)."
}

Failure boundary

BREAK_CLAIMED · SELF_FLAGGED

The input a=0 makes the quadratic-formula denominator 2a equal zero. A robust solver must reject it as nonquadratic or explicitly fall back to a linear solver; blindly applying the quadratic formula produces an undefined result.

Other records

Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

• VERIFIED_LEDGER.json
• evidence-tranche6/tranche6-results.json
• BOUNDARY_HAZARD_CATALOG.json