{
  "capability_id": "math.quadratic",
  "tool_id": "quadratic",
  "engine": "quadratic",
  "calculation_version": "1.1.0",
  "generated_at": "2026-09-20T09:01:32.955Z",
  "engine_build": "2026-09-19",
  "test_count": 9,
  "passed": 9,
  "failed": 0,
  "tests_summary": {
    "golden": {
      "total": 6,
      "passed": 6
    },
    "boundary": {
      "total": 3,
      "passed": 3
    },
    "property": {
      "total": 0,
      "passed": 0
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  "engine_tested": true,
  "expert_reviewed": false,
  "internally_reviewed": false,
  "review_kind": null,
  "source_checked": true,
  "reviewed_by": null,
  "reviewer_role": null,
  "last_reviewed": null,
  "expert_review": {
    "status": "optional_not_performed",
    "reviewer": null,
    "scope": [
      "formula",
      "input semantics",
      "assumptions",
      "limitations",
      "sources",
      "tests"
    ],
    "note": "Domain expert review is optional under CVP (CVP-EXP-01). Absence does not block CVP Verified. See Review policy for supplemental domain assurance."
  },
  "internal_review": {
    "status": "not_applicable",
    "reviewer": null
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  "page": "https://www.calculatorx.com/calc/math/quadratic",
  "api": "https://www.calculatorx.com/api/v1/calc/quadratic",
  "schema_input": "https://www.calculatorx.com/schemas/math.quadratic.input.json",
  "schema_output": "https://www.calculatorx.com/schemas/math.quadratic.output.json",
  "assumptions": [
    "a ≠ 0; coefficients are real numbers (fractions allowed)."
  ],
  "limitations": [
    "a = 0 → NOT_QUADRATIC",
    "Missing coefficient → MISSING_REQUIRED_INPUT",
    "Δ < 0 equation yields complex conjugate roots {real, imag}",
    "Inequality is mode=inequality; discovery /quadratic-inequality",
    "Not cubic, not CAS factoring"
  ],
  "boundary_conditions": [
    "a = 0 → rejected as not quadratic"
  ],
  "sources": [
    {
      "title": "OpenStax Intermediate Algebra 2e — Quadratic Formula",
      "section": "9.3 x = (−b ± √(b² − 4ac)) / (2a)",
      "url": "https://openstax.org/books/intermediate-algebra-2e/pages/9-3-solve-quadratic-equations-using-the-quadratic-formula",
      "supports": "Quadratic formula and discriminant classification of the roots of ax² + bx + c = 0.",
      "kind": "reference"
    },
    {
      "title": "OpenStax College Algebra 2e — Quadratic Functions",
      "section": "5.1 Vertex and axis of y = ax² + bx + c",
      "url": "https://openstax.org/books/college-algebra-2e/pages/5-1-quadratic-functions",
      "supports": "Vertex / axis of symmetry of y = ax²+bx+c",
      "kind": "reference"
    }
  ],
  "methods": [],
  "fingerprint": "sha256:296cd3ea369b5bab14dd757dc3e0665d63485936e04bee702b8a48a76a9dafeb",
  "tests": [
    {
      "id": "two-real",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "a": 1,
        "b": -6,
        "c": 8
      },
      "expect": {
        "x1": 4,
        "x2": 2,
        "delta": 4
      },
      "expectWarning": null,
      "actual": {
        "a": 1,
        "b": -6,
        "c": 8,
        "mode": "solve",
        "delta": 4,
        "xv": 3,
        "yv": -1,
        "axis": 3,
        "y_intercept": 8,
        "opens": "up",
        "opens_label": "Upward",
        "root_type": "two_real",
        "mode_label": "Two distinct real roots",
        "formula": "x = (−b ± √Δ)/(2a)",
        "heading": "Roots",
        "explainer": "Δ > 0 → two distinct real roots.",
        "equation": "x² − 6x + 8 = 0",
        "inputs_summary": "a=1 · b=-6 · c=8",
        "calculation_version": "1.1.0",
        "x1": 4,
        "x2": 2,
        "value": "4, 2",
        "display": {
          "equation": "x² − 6x + 8 = 0",
          "delta": "4",
          "x1": "4",
          "x2": "2",
          "roots": "4, 2",
          "vertex": "(3, −1)",
          "vertex_x": "3",
          "vertex_y": "−1",
          "axis": "3",
          "y_intercept": "8",
          "opens": "Upward",
          "factorization": "(x − 2)(x − 4) = 0",
          "exact": "",
          "equation_lhs": "x² − 6x + 8"
        },
        "steps": [
          "x² − 6x + 8 = 0",
          "",
          "Δ = b² − 4ac",
          "  = (−6)² − 4(1)(8)",
          "  = 36 − 32",
          "  = 4",
          "",
          "x = (−b ± √Δ) / 2a",
          "  = (6 ± √4) / 2",
          "x₁ = 4",
          "x₂ = 2",
          "",
          "Vertex:",
          "xᵥ = −b / 2a = 3",
          "yᵥ = −1",
          "",
          "Axis: x = 3",
          "",
          "Factor form: (x − 2)(x − 4) = 0"
        ]
      },
      "tol": 1e-9
    },
    {
      "id": "repeated",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "a": 1,
        "b": -4,
        "c": 4
      },
      "expect": {
        "x1": 2,
        "x2": 2,
        "delta": 0
      },
      "expectWarning": null,
      "actual": {
        "a": 1,
        "b": -4,
        "c": 4,
        "mode": "solve",
        "delta": 0,
        "xv": 2,
        "yv": 0,
        "axis": 2,
        "y_intercept": 4,
        "opens": "up",
        "opens_label": "Upward",
        "root_type": "repeated",
        "mode_label": "One repeated real root",
        "formula": "x = (−b ± √Δ)/(2a)",
        "heading": "Roots",
        "explainer": "Δ = 0 → one repeated real root.",
        "equation": "x² − 4x + 4 = 0",
        "inputs_summary": "a=1 · b=-4 · c=4",
        "calculation_version": "1.1.0",
        "x1": 2,
        "x2": 2,
        "value": "2",
        "display": {
          "equation": "x² − 4x + 4 = 0",
          "delta": "0",
          "x1": "2",
          "x2": "2",
          "roots": "2",
          "vertex": "(2, 0)",
          "vertex_x": "2",
          "vertex_y": "0",
          "axis": "2",
          "y_intercept": "4",
          "opens": "Upward",
          "factorization": "(x − 2)² = 0",
          "exact": "",
          "equation_lhs": "x² − 4x + 4"
        },
        "steps": [
          "x² − 4x + 4 = 0",
          "",
          "Δ = b² − 4ac",
          "  = (−4)² − 4(1)(4)",
          "  = 16 − 16",
          "  = 0",
          "",
          "x = −b / 2a",
          "  = 2",
          "",
          "Vertex:",
          "xᵥ = −b / 2a = 2",
          "yᵥ = 0",
          "",
          "Axis: x = 2",
          "",
          "Factor form: (x − 2)² = 0"
        ]
      },
      "tol": 1e-9
    },
    {
      "id": "complex",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "a": 1,
        "b": 2,
        "c": 5
      },
      "expect": {
        "real": -1,
        "imag": 2,
        "delta": -16
      },
      "expectWarning": null,
      "actual": {
        "a": 1,
        "b": 2,
        "c": 5,
        "mode": "solve",
        "delta": -16,
        "xv": -1,
        "yv": 4,
        "axis": -1,
        "y_intercept": 5,
        "opens": "up",
        "opens_label": "Upward",
        "root_type": "complex",
        "mode_label": "Two complex conjugate roots",
        "formula": "x = (−b ± √Δ)/(2a)",
        "heading": "Roots",
        "explainer": "Δ < 0 → two complex conjugate roots.",
        "equation": "x² + 2x + 5 = 0",
        "inputs_summary": "a=1 · b=2 · c=5",
        "calculation_version": "1.1.0",
        "real": -1,
        "imag": 2,
        "value": "-1 ± 2i",
        "display": {
          "equation": "x² + 2x + 5 = 0",
          "delta": "−16",
          "x1": "−1 + 2i",
          "x2": "−1 − 2i",
          "roots": "−1 ± 2i",
          "vertex": "(−1, 4)",
          "vertex_x": "−1",
          "vertex_y": "4",
          "axis": "−1",
          "y_intercept": "5",
          "opens": "Upward",
          "factorization": "",
          "exact": "",
          "equation_lhs": "x² + 2x + 5"
        },
        "steps": [
          "x² + 2x + 5 = 0",
          "",
          "Δ = b² − 4ac",
          "  = 2² − 4(1)(5)",
          "  = 4 − 20",
          "  = −16",
          "",
          "x = (−b ± √|Δ| i) / 2a",
          "  = −1 ± 2i",
          "",
          "Vertex:",
          "xᵥ = −b / 2a = −1",
          "yᵥ = 4",
          "",
          "Axis: x = −1"
        ]
      },
      "tol": 1e-9
    },
    {
      "id": "default-two-real",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "a": 1,
        "b": -5,
        "c": 6
      },
      "expect": {
        "x1": 3,
        "x2": 2,
        "delta": 1
      },
      "expectWarning": null,
      "actual": {
        "a": 1,
        "b": -5,
        "c": 6,
        "mode": "solve",
        "delta": 1,
        "xv": 2.5,
        "yv": -0.25,
        "axis": 2.5,
        "y_intercept": 6,
        "opens": "up",
        "opens_label": "Upward",
        "root_type": "two_real",
        "mode_label": "Two distinct real roots",
        "formula": "x = (−b ± √Δ)/(2a)",
        "heading": "Roots",
        "explainer": "Δ > 0 → two distinct real roots.",
        "equation": "x² − 5x + 6 = 0",
        "inputs_summary": "a=1 · b=-5 · c=6",
        "calculation_version": "1.1.0",
        "x1": 3,
        "x2": 2,
        "value": "3, 2",
        "display": {
          "equation": "x² − 5x + 6 = 0",
          "delta": "1",
          "x1": "3",
          "x2": "2",
          "roots": "3, 2",
          "vertex": "(2.5, −0.25)",
          "vertex_x": "2.5",
          "vertex_y": "−0.25",
          "axis": "2.5",
          "y_intercept": "6",
          "opens": "Upward",
          "factorization": "(x − 2)(x − 3) = 0",
          "exact": "",
          "equation_lhs": "x² − 5x + 6"
        },
        "steps": [
          "x² − 5x + 6 = 0",
          "",
          "Δ = b² − 4ac",
          "  = (−5)² − 4(1)(6)",
          "  = 25 − 24",
          "  = 1",
          "",
          "x = (−b ± √Δ) / 2a",
          "  = (5 ± √1) / 2",
          "x₁ = 3",
          "x₂ = 2",
          "",
          "Vertex:",
          "xᵥ = −b / 2a = 2.5",
          "yᵥ = −0.25",
          "",
          "Axis: x = 2.5",
          "",
          "Factor form: (x − 2)(x − 3) = 0"
        ]
      },
      "tol": 1e-9
    },
    {
      "id": "irrational",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "a": 1,
        "b": -8,
        "c": 5
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      "expect": {
        "delta": 44
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      "expectWarning": null,
      "actual": {
        "a": 1,
        "b": -8,
        "c": 5,
        "mode": "solve",
        "delta": 44,
        "xv": 4,
        "yv": -11,
        "axis": 4,
        "y_intercept": 5,
        "opens": "up",
        "opens_label": "Upward",
        "root_type": "two_real",
        "mode_label": "Two distinct real roots",
        "formula": "x = (−b ± √Δ)/(2a)",
        "heading": "Roots",
        "explainer": "Δ > 0 → two distinct real roots.",
        "equation": "x² − 8x + 5 = 0",
        "inputs_summary": "a=1 · b=-8 · c=5",
        "calculation_version": "1.1.0",
        "x1": 7.3166247903554,
        "x2": 0.6833752096446002,
        "value": "7.31662479, 0.6833752096",
        "display": {
          "equation": "x² − 8x + 5 = 0",
          "delta": "44",
          "x1": "7.31662479",
          "x2": "0.6833752096",
          "roots": "7.31662479, 0.6833752096",
          "vertex": "(4, −11)",
          "vertex_x": "4",
          "vertex_y": "−11",
          "axis": "4",
          "y_intercept": "5",
          "opens": "Upward",
          "factorization": "",
          "exact": "4 ± √11",
          "equation_lhs": "x² − 8x + 5"
        },
        "steps": [
          "x² − 8x + 5 = 0",
          "",
          "Δ = b² − 4ac",
          "  = (−8)² − 4(1)(5)",
          "  = 64 − 20",
          "  = 44",
          "",
          "x = (−b ± √Δ) / 2a",
          "  = (8 ± √44) / 2",
          "  = 4 ± √11",
          "x₁ = 7.31662479",
          "x₂ = 0.6833752096",
          "",
          "Vertex:",
          "xᵥ = −b / 2a = 4",
          "yᵥ = −11",
          "",
          "Axis: x = 4"
        ]
      },
      "tol": 1e-9
    },
    {
      "id": "not-quadratic",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "a": 0,
        "b": 2,
        "c": 1
      },
      "expectError": "NOT_QUADRATIC",
      "code": "NOT_QUADRATIC",
      "error": "a must be nonzero. If a = 0 the equation is linear, not quadratic."
    },
    {
      "id": "cancellation",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "a": 1,
        "b": -100000000,
        "c": 1
      },
      "expect": {
        "x2": 1e-8
      },
      "expectWarning": null,
      "actual": {
        "a": 1,
        "b": -100000000,
        "c": 1,
        "mode": "solve",
        "delta": 9999999999999996,
        "xv": 50000000,
        "yv": -2499999999999999,
        "axis": 50000000,
        "y_intercept": 1,
        "opens": "up",
        "opens_label": "Upward",
        "root_type": "two_real",
        "mode_label": "Two distinct real roots",
        "formula": "x = (−b ± √Δ)/(2a)",
        "heading": "Roots",
        "explainer": "Δ > 0 → two distinct real roots.",
        "equation": "x² − 100000000x + 1 = 0",
        "inputs_summary": "a=1 · b=-100000000 · c=1",
        "calculation_version": "1.1.0",
        "x1": 100000000,
        "x2": 1e-8,
        "value": "100000000, 1e-8",
        "display": {
          "equation": "x² − 100000000x + 1 = 0",
          "delta": "9999999999999996",
          "x1": "100000000",
          "x2": "1e−8",
          "roots": "100000000, 1e−8",
          "vertex": "(50000000, −2499999999999999)",
          "vertex_x": "50000000",
          "vertex_y": "−2499999999999999",
          "axis": "50000000",
          "y_intercept": "1",
          "opens": "Upward",
          "factorization": "",
          "exact": "50000000 ± √2499999999999999",
          "equation_lhs": "x² − 100000000x + 1"
        },
        "steps": [
          "x² − 100000000x + 1 = 0",
          "",
          "Δ = b² − 4ac",
          "  = (−100000000)² − 4(1)(1)",
          "  = 10000000000000000 − 4",
          "  = 9999999999999996",
          "",
          "x = (−b ± √Δ) / 2a",
          "  = (100000000 ± √9999999999999996) / 2",
          "  = 50000000 ± √2499999999999999",
          "x₁ = 100000000",
          "x₂ = 1e−8",
          "",
          "Vertex:",
          "xᵥ = −b / 2a = 50000000",
          "yᵥ = −2499999999999999",
          "",
          "Axis: x = 50000000"
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      "tol": 0.000001,
      "tol_type": "relative"
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    {
      "id": "missing-a",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "b": 1,
        "c": 1
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      "expectError": "MISSING_REQUIRED_INPUT",
      "code": "MISSING_REQUIRED_INPUT",
      "error": "Enter a finite quadratic coefficient a."
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      "id": "boundary-quad-a0",
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      "inputs": {
        "a": 0,
        "b": 1,
        "c": 1
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      "expectError": "NOT_QUADRATIC",
      "code": "NOT_QUADRATIC",
      "error": "a must be nonzero. If a = 0 the equation is linear, not quadratic."
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