{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-quadratic-equation",
  "semantic_family_id": "sf-quadratic-equation",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "一元實二次方程式"
  },
  "summary": {
    "zh-Hant": "對非零二次係數的實方程，用判別式區分實根數，將方程式、根集合與重根分開。"
  },
  "statement": {
    "zh-Hant": "在實數論域中，令 q(x)=ax²+bx+c，a、b、c∈R 且 a≠0，則 q(x)=0 是一元二次方程式。令 D=b²−4ac：D<0 時無實根，D=0 時只有一個相異實根，D>0 時有兩個相異實根；D≥0 時根為 (-b±√D)/(2a)。"
  },
  "formula": {
    "tex": "q(x)=0",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>q</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "q",
          "glyph": "q"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "x",
            "glyph": "x"
          }
        ]
      },
      "rhs": {
        "type": "number",
        "value": 0
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "b386dbef4eee41808a1cadde79805dba44b355f1d9188e2a5f49639e0c64b118"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "論域固定為 R，a≠0；a=0 的方程不屬於二次方程式。"
    },
    {
      "id": "a2",
      "zh-Hant": "主平方根只在 D≥0 時使用；根集合重複值只列一次，不把重數當相異根數。"
    }
  ],
  "symbols": [
    {
      "token": "q",
      "role": "quadratic_polynomial",
      "role_zh": "二次多項式",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "real_unknown",
      "role_zh": "實數未知數",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-equation",
      "title_zh": "方程式與等式",
      "reason_zh": "需先固定未知數論域及解的代入條件。"
    },
    {
      "id": "mko-power-root",
      "title_zh": "自然數冪與主平方根",
      "reason_zh": "二次冪及 D≥0 的主平方根支撐根公式。"
    },
    {
      "id": "mko-polynomial",
      "title_zh": "一元有理係數多項式",
      "reason_zh": "明示整數係數題目是實二次多項式的精確子類。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "先檢查 a≠0，再算判別式。D 的符號分類僅針對實根；不能把 D<0 的無實根寫成所有數域都無解。",
      "在 a≠0 的前提下，平方完成給出 q(x)=0 ⇔ (2ax+b)²=D；D≥0 時可解回根公式。D=0 的兩個公式分支相同；根集合只列一個值。",
      "題庫使用整數係數、候選有理根的分母清除代入，以及非負完全平方 D 的精確根集合。不使用浮點近似比較根，也不宣稱全部實數平方根的形式證明已完成。"
    ],
    "common_misconception_zh": "略過 a≠0、把 D=0 說成兩個相異根，或把無實根誤當無複數根。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "指定方程式的判別式與有理根可精確重算；一般根公式與實根分類仍未附形式 Evidence。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "綁定實數非零二次係數、平方完成等價、判別式分類和精確根集合。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-quadratic-equation.md 整理；2026-10-03 脈衝 R04，英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-03"
  }
}