{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261003-r04-quadratic-equation-0043",
  "pilot_number": "R0055-28418",
  "number": 28418,
  "family_id": "qf-quadratic-discriminant",
  "title": {
    "zh-Hant": "第 28418 題：判別式與相異實根數",
    "en": "Question 28418: Discriminant and Distinct Real Roots"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261003-r04-quadratic-equation-0043",
    "batch_id": "batch-20261003-r04-quadratic-equation-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-quadratic-equation"
    ],
    "title_zh": "判別式與相異實根數",
    "stem_zh": "在實數論域，(-1)x²+(5)x+(6)=0 的 a≠0。求判別式 D 及相異實根數。",
    "answer": {
      "kind": "short_text",
      "value": "D=49；相異實根數=2",
      "explanation_zh": "D=49；相異實根數=2。D=b²−4ac；負、零、正分別給零、一、兩個相異實根，不把重根算兩次。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 3,
      "skills": [
        "判別式與相異實根數"
      ]
    },
    "variant": {
      "family_id": "qf-quadratic-discriminant",
      "seed": 202610034043,
      "parameters": {
        "a": -1,
        "b": 5,
        "c": 6,
        "domain": "reals",
        "operation": "discriminant",
        "semantic_template": "qf-quadratic-discriminant/{\"a\":-1,\"b\":5,\"c\":6,\"domain\":\"reals\",\"operation\":\"discriminant\"}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication"
      ],
      "warning_zh": "僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "Codex independent peer (inherited model; exact runtime model not exposed)",
      "generated_at": "2026-10-03",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261003-r04-quadratic-equation-0043",
    "batch_id": "batch-20261003-r04-quadratic-equation-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "3bf3b6cf790560e00dfde2012d827396520e091f594528036ac72c4784db1a79",
      "family_id": "qf-quadratic-discriminant",
      "seed": 202610034043,
      "semantic_template": "qf-quadratic-discriminant/{\"a\":-1,\"b\":5,\"c\":6,\"domain\":\"reals\",\"operation\":\"discriminant\"}"
    },
    "fields": {
      "title": "Discriminant and Distinct Real Roots",
      "stem": "Over the reals, (-1)x²+(5)x+(6)=0 has a≠0. Find the discriminant D and the number of distinct real roots.",
      "answer": {
        "kind": "short_text",
        "value": "D=49; distinct real roots=2",
        "explanation": "D=49; distinct real roots=2. Compute D=b²−4ac; negative, zero, and positive D give zero, one, and two distinct real roots respectively. A repeated root is counted once."
      },
      "skills": [
        "Discriminant and Distinct Real Roots"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "在實數論域，(-1)x²+(5)x+(6)=0 的 a≠0。求判別式 D 及相異實根數。",
        "en": "Over the reals, (-1)x²+(5)x+(6)=0 has a≠0. Find the discriminant D and the number of distinct real roots."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [qf-quadratic-discriminant, discriminant]",
      "after": "derived = [[discriminant_and_number_of_DISTINCT_real_roots]]",
      "relation": "implies",
      "rule_id": "source_answer_roles_or_ordered_fields_without_JS",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此頁明列本題scalar／label／set／point或mode欄序，不強套現代tuple。舊題domain與operation底線格式按原資料，分數整數亦保/1；原中英快照不改。",
        "en": "List this scalar/label/set/point answer role or mode-specific ordered fields,not an imposed universal tuple.Retain legacy domain/underscore operation and integer fractions with/1;original bilingual snapshots stay unchanged."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。",
          "en": "Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "Legacy quadratic ORIGINAL b squared := mul(5, 5)",
      "after": "Legacy quadratic ORIGINAL b squared = 25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Legacy quadratic ORIGINAL b squared，不以小數近似替代。",
          "en": "Recompute Legacy quadratic ORIGINAL b squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "5",
          "5"
        ],
        "output": "25",
        "label": "Legacy quadratic ORIGINAL b squared"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Legacy quadratic ORIGINAL a times c := mul(-1, 6)",
      "after": "Legacy quadratic ORIGINAL a times c = -6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Legacy quadratic ORIGINAL a times c，不以小數近似替代。",
          "en": "Recompute Legacy quadratic ORIGINAL a times c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "6"
        ],
        "output": "-6",
        "label": "Legacy quadratic ORIGINAL a times c"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "Legacy quadratic ORIGINAL four times ac := mul(4, -6)",
      "after": "Legacy quadratic ORIGINAL four times ac = -24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Legacy quadratic ORIGINAL four times ac，不以小數近似替代。",
          "en": "Recompute Legacy quadratic ORIGINAL four times ac with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "-6"
        ],
        "output": "-24",
        "label": "Legacy quadratic ORIGINAL four times ac"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Legacy quadratic ORIGINAL b squared minus fourac := sub(25, -24)",
      "after": "Legacy quadratic ORIGINAL b squared minus fourac = 49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Legacy quadratic ORIGINAL b squared minus fourac，不以小數近似替代。",
          "en": "Recompute Legacy quadratic ORIGINAL b squared minus fourac with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "25",
          "-24"
        ],
        "output": "49",
        "label": "Legacy quadratic ORIGINAL b squared minus fourac"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "source roles = [-1, 5, 6, 49]",
      "after": "derived = [4a*(a*x^2+b*x+c)=(2a*x+b)^2-D, 2]",
      "relation": "implies",
      "rule_id": "Legacy_quadratic_GENERAL_complete_square_iff_DISTINCT_real_roots",
      "conditions": [],
      "explanation": {
        "zh-Hant": "a非0使4a可逆，方程等價(2ax+b)²=D。D<0無實根，D0一個值，D>0兩個相異值；重根只算一次，D為不設衍生SAFE上限的BigInt。",
        "en": "With a!=0,multiplication by4a is invertible and the equation is equivalent to(2ax+b)²=D.Negative D gives no real root,zero D one value and positive D two DISTINCT values.A repeated root counts once;D is unbounded derived BigInt."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。",
          "en": "Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": "D=49；相異實根數=2"
  },
  "conclusion": {
    "canonical_value": "D=49；相異實根數=2",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261003-r04-quadratic-equation-foundations",
      "batch_path": "public/data/questions/batches/batch-20261003-r04-quadratic-equation-foundations/questions.jsonl",
      "batch_sha256": "66a176b962ca84e02bebfdfca16c81ee9677788154f1765c2bfa965a41fce5f6",
      "original_line": 43,
      "original_line_sha256": "7d7815a50e3487052b19e0e1fc9fd6038bfbd2e29504ef6efa4883de9e1a0622",
      "question_compact_sha256": "3bf3b6cf790560e00dfde2012d827396520e091f594528036ac72c4784db1a79",
      "english_row_compact_sha256": "98d7bd56c701bcec444c924ca6fac4bdee69d61b30cf041c4579685fe71b006c",
      "english_manifest_path": "public/data/i18n/editions/batch-20261003-r04-quadratic-equation-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261003-r04-quadratic-equation-foundations.en.jsonl"
    },
    "mko_bindings": {
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        "original_path": "public/data/mko/quadratic-equation.json",
        "english_path": "public/data/i18n/editions/mko-quadratic-equation.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "b5763df142e86ef7f9787b6ea86050afc60bd3c5c53538d4839606827cc44686",
    "original_candidate_sha256": "0e69adbb8e7479bb91625ef17f1229e36cb4118ebd74c4a63b314dcfb9b976d7",
    "original_candidate_line_sha256": "0e69adbb8e7479bb91625ef17f1229e36cb4118ebd74c4a63b314dcfb9b976d7"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-010-p56",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "6af367dae24a584f3596525c4953bbc2115343efd68cc9083693eb5c1ef2e683",
      "question_first_witness_sha256": "611eb69f3716404b968296a8b0d7d8c61b5ca18129d406f1ff9c7c11e10ffb4e"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 6,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "83dd743889ef4596b44d78d36678c1b964f9ac8cd40dd9ba27a0a6db7166d652",
      "arithmetic_steps": 4
    },
    "uses_answer_as_premise": false,
    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}