Source: https://guthlabs.ai/docs/instruments/quadratic-equation-solver/mechanism/

instrument mechanism / verified-clean

# Quadratic Equation Solver mechanism

The inspectable method record for Quadratic Equation Solver, kept separate from implementation and qualification.

Status: **VERIFIED_CLEAN**Lifecycle: **verified-clean**

## 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

[01**Overview**](<https://guthlabs.ai/docs/instruments/quadratic-equation-solver/>)[03**Implementation**](<https://guthlabs.ai/docs/instruments/quadratic-equation-solver/implementation/>)[04**Qualification**](<https://guthlabs.ai/docs/instruments/quadratic-equation-solver/qualification/>)

## 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`