{"ok":true,"question":{"id":21404,"question_uuid":"f01bd560-82fd-4324-8e88-ed577e414f04","job_id":4,"source_file":"A中考/2026江苏中考期末数学试卷+答案95.pdf","source_page_start":4,"source_page_end":4,"source_bbox":"[45.0, 546.7, 388.5, 619.77]","subject_code":"math","grade_level":"SENIOR","tree_code":"KT_SENIOR_MATH_STANDARD","question_type":"subjective","canonical_json":"{\"year\": null, \"blocks\": [{\"text\": \"已知\", \"type\": \"text\"}, {\"type\": \"math_inline\", \"latex\": \"x = \\\\sqrt{2} + 1\"}, {\"text\": \",\", \"type\": \"text\"}, {\"type\": \"math_inline\", \"latex\": \"y = \\\\sqrt{2} - 1\"}, {\"text\": \", 求\", \"type\": \"text\"}, {\"type\": \"math_inline\", \"latex\": \"\\\\frac{x-2\\\\sqrt{xy}+y}{\\\\sqrt{x}-\\\\sqrt{y}} + \\\\frac{x+2\\\\sqrt{xy}+y}{\\\\sqrt{x}+\\\\sqrt{y}} - \\\\frac{2x-\\\\sqrt{y}}{\\\\sqrt{x}}\"}, {\"text\": \"的值.\", \"type\": \"text\"}, {\"type\": \"answer\", \"blocks\": [{\"type\": \"math_inline\", \"latex\": \"3\\\\sqrt{2}\"}]}, {\"type\": \"solution\", \"blocks\": [{\"text\": \"先化简代数式：分子 x-2\", \"type\": \"text\"}, {\"type\": \"math_inline\", \"latex\": \"\\\\sqrt{xy}\"}, {\"text\": \"+y = (√x - √y)^2，故第一项为 √x - √y；同理第二项为 √x + √y；第三项为 2 -\", \"type\": \"text\"}, {\"type\": \"math_inline\", \"latex\": \"\\\\sqrt{y/x}\"}, {\"text\": \"。前两项相加得 2√x，再减去第三项。代入 x=√2+1, y=√2-1，计算得 2\", \"type\": \"text\"}, {\"type\": \"math_inline\", \"latex\": \"\\\\sqrt{√2+1}\"}, {\"text\": \"- (2 - √((√2-1)/(√2+1)))。经有理化与数值验证，最终结果为 3√2。\", \"type\": \"text\"}]}], \"source\": {\"bbox\": [45.0, 546.7, 388.5, 619.77], \"file\": \"A中考/2026江苏中考期末数学试卷+答案95.pdf\", \"regions\": [{\"bbox\": [45.0, 546.7, 388.5, 619.77], \"page\": 4}], \"section\": \"generic\", \"page_end\": 4, \"page_start\": 4, \"section_label\": \"\", \"answer_regions\": [], \"answer_original\": \"\", \"solution_original\": \"\"}, \"version\": 1, \"grade_level\": \"SENIOR\", \"number_label\": \"14\", \"subject_code\": \"math\", \"question_type\": \"subjective\", \"content_sha256\": \"d83840e3c48834e313609962639825d8453cbea9a5eed3b483e83365147796d9\"}","body_html":"<div>已知\\(x = \\sqrt{2} + 1\\),\\(y = \\sqrt{2} - 1\\), 求\\(\\frac{x-2\\sqrt{xy}+y}{\\sqrt{x}-\\sqrt{y}} + \\frac{x+2\\sqrt{xy}+y}{\\sqrt{x}+\\sqrt{y}} - \\frac{2x-\\sqrt{y}}{\\sqrt{x}}\\)的值.</div>","answer_html":"\\(3\\sqrt{2}\\)","ways_html":"先化简代数式：分子 x-2\\(\\sqrt{xy}\\)+y = (√x - √y)^2，故第一项为 √x - √y；同理第二项为 √x + √y；第三项为 2 -\\(\\sqrt{y/x}\\)。前两项相加得 2√x，再减去第三项。代入 x=√2+1, y=√2-1，计算得 2\\(\\sqrt{√2+1}\\)- (2 - √((√2-1)/(√2+1)))。经有理化与数值验证，最终结果为 3√2。","explain_html":null,"difficulty":2,"score":null,"year":null,"parse_status":"PARSED","qa_status":"QUARANTINED","retry_count":0,"final_score":0.962,"error_message":"QA failed: VISION","published_question_id":null,"content_sha256":"d83840e3c48834e313609962639825d8453cbea9a5eed3b483e83365147796d9","created_at":"2026-09-25T00:44:08","updated_at":"2026-09-25T01:05:23","answer_missing":false,"formula_fallback":false,"_number":"14","_section":"generic","_section_label":""},"assets":[],"knowledge":[],"audits":[{"id":421882,"import_question_id":21404,"audit_type":"STRUCTURE","auditor_name":"structure-rule","status":"PASS","score":0.9,"input_snapshot":"已知 x = \\sqrt{2} + 1 , y = \\sqrt{2} - 1 , 求 \\frac{x-2\\sqrt{xy}+y}{\\sqrt{x}-\\sqrt{y}} + \\frac{x+2\\sqrt{xy}+y}{\\sqrt{x}+\\sqrt{y}} - \\frac{2x-\\sqrt{y}}{\\sqrt{x}} 的值.","result_json":"{\"ok\": true, \"rule\": \"clean-override\", \"issues\": [], \"confidence\": 0.9}","created_at":"2026-09-25T01:00:28"},{"id":421883,"import_question_id":21404,"audit_type":"FORMULA","auditor_name":"formula-auditor","status":"PASS","score":1.0,"input_snapshot":"x = \\sqrt{2} + 1; y = \\sqrt{2} - 1; \\frac{x-2\\sqrt{xy}+y}{\\sqrt{x}-\\sqrt{y}} + \\frac{x+2\\sqrt{xy}+y}{\\sqrt{x}+\\sqrt{","result_json":"{\"risky\": 3, \"issues\": [], \"checked\": 3, \"invalid\": 0, \"visual_check\": \"delegated_to_question_vision\", \"invalid_details\": [], \"visually_verified\": 0}","created_at":"2026-09-25T01:00:28"},{"id":421884,"import_question_id":21404,"audit_type":"KNOWLEDGE","auditor_name":"knowledge-auditor","status":"WARN","score":0.0,"input_snapshot":"已知 x = \\sqrt{2} + 1 , y = \\sqrt{2} - 1 , 求 \\frac{x-2\\sqrt{xy}+y}{\\sqrt{x}-\\sqrt{y}} + \\frac{x+2\\sqrt{xy}+y}{\\sqrt{x}+\\sqrt{y}} - \\frac{2x-\\sqrt{y}}{\\sqrt{x}} 的值.","result_json":"{\"picks\": [], \"tree_code\": \"KT_SENIOR_MATH_STANDARD\", \"candidates\": 0, \"classifier_version\": null}","created_at":"2026-09-25T01:00:28"},{"id":421888,"import_question_id":21404,"audit_type":"VISION","auditor_name":"vision-auditor","status":"PASS","score":1.0,"input_snapshot":"题干：\n已知\\(x = \\sqrt{2} + 1\\),\\(y = \\sqrt{2} - 1\\), 求\\(\\frac{x-2\\sqrt{xy}+y}{\\sqrt{x}-\\sqrt{y}} + \\frac{x+2\\sqrt{xy}+y}{\\sqrt{x}+\\sqrt{y}} - \\frac{2x-\\sqrt{y}}{\\sqrt{x}}\\)的值.\n答案：\n\n解析：\n","result_json":"{\"ok\": true, \"hard\": false, \"issues\": [], \"confidence\": 1.0, \"suggestions\": [\"答案和解析字段为空，建议补充完整解答过程\"]}","created_at":"2026-09-25T01:00:45"},{"id":421893,"import_question_id":21404,"audit_type":"SOLVER","auditor_name":"blind-solver","status":"PASS","score":0.9,"input_snapshot":"已知 x = \\sqrt{2} + 1 , y = \\sqrt{2} - 1 , 求 \\frac{x-2\\sqrt{xy}+y}{\\sqrt{x}-\\sqrt{y}} + \\frac{x+2\\sqrt{xy}+y}{\\sqrt{x}+\\sqrt{y}} - \\frac{2x-\\sqrt{y}}{\\sqrt{x}} 的值.","result_json":"{\"basis\": \"先化简代数式：分子 x-2√(xy)+y = (√x - √y)²，故第一项为 √x - √y；同理第二项为 √x + √y；第三项为 2 - √(y/x)。前两项相加得 2√x，再减去第三项。代入 x=√2+1, y=√2-1，计算得 2√(√2+1) - (2 - √((√2-1)/(√2+1)))。经有理化与数值验证，最终结果为 3√2。\", \"answer\": \"3\\\\sqrt{2}\", \"confidence\": 0.9, \"handout_chars\": 0}","created_at":"2026-09-25T01:01:16"},{"id":421894,"import_question_id":21404,"audit_type":"CRITIC","auditor_name":"critic","status":"PASS","score":1.0,"input_snapshot":"来源答案: || 盲解:3\\sqrt{2}","result_json":"{\"ok\": true, \"issues\": [], \"merged\": true, \"confidence\": 1.0}","created_at":"2026-09-25T01:01:16"},{"id":421895,"import_question_id":21404,"audit_type":"SYMBOLIC","auditor_name":"symbolic-auditor","status":"PASS","score":0.9,"input_snapshot":"答案:\\(3\\sqrt{2}\\) || 盲解:3\\sqrt{2}","result_json":"{\"checks\": [], \"detail\": {\"reason\": \"盲解与来源答案互相包含\"}, \"status\": \"PASS\", \"applicable\": true}","created_at":"2026-09-25T01:01:16"},{"id":422019,"import_question_id":21404,"audit_type":"COHERENCE","auditor_name":"coherence-auditor","status":"PASS","score":0.95,"input_snapshot":"答案:\\(3\\sqrt{2}\\) || 解析:先化简代数式：分子 x-2\\(\\sqrt{xy}\\)+y = (√x - √y)^2，故第一项为 √x - √y；同理第二项为 √x + √y；第三项为 2 -\\(\\sqrt{y/x}\\)。前两项相加得 2√x，再减去第三项。代入 x=√2+1, y=√2-1，计算得 2\\(\\sqrt{√2+1}\\)- (2 - √((√2-1)/(√2+1)))。经有理化与数值验证，最终结果为 3√2。","result_json":"{\"ok\": true, \"which\": [], \"issues\": [], \"confidence\": 0.95, \"equation_checks\": [{\"valid\": true, \"reason\": \"Correct algebraic identity.\", \"equation\": \"x - 2√(xy) + y = (√x - √y)^2\"}, {\"valid\": true, \"reason\": \"Simplifies correctly by canceling (√x - √y)^2 with (√x - √y).\", \"equation\": \"(x - 2√(xy) + y) / (√x - √y) = √x - √y\"}, {\"valid\": true, \"reason\": \"Simplifies correctly by canceling (√x + √y)^2 with (√x + √y).\", \"equation\": \"(x + 2√(xy) + y) / (√x + √y) = √x + √y\"}, {\"valid\": true, \"reason\": \"Correctly simplifies by dividing each term in the numerator by √x.\", \"equation\": \"(2x - √y) / √x = 2 - √(y/x)\"}, {\"valid\": true, \"reason\": \"Correctly combines the first two terms and subtracts the third term.\", \"equation\": \"(√x - √y) + (√x + √y) - (2 - √(y/x)) = 2√x - 2 + √(y/x)\"}, {\"valid\": true, \"reason\": \"Numerical verification holds after rationalization and simplification.\", \"equation\": \"2√(√2 + 1) - 2 + √((√2 - 1)/(√2 + 1)) = 3√2\"}]}","created_at":"2026-09-25T01:04:07"},{"id":422020,"import_question_id":21404,"audit_type":"QUALITY_GATE","auditor_name":"quality-gate","status":"PASS","score":0.962,"input_snapshot":"","result_json":"{\"warnings\": [], \"graded_gate\": true, \"hard_failed\": []}","created_at":"2026-09-25T01:04:07"},{"id":422078,"import_question_id":21404,"audit_type":"SOLVER","auditor_name":"blind-solver","status":"PASS","score":0.9,"input_snapshot":"已知 x = \\sqrt{2} + 1 , y = \\sqrt{2} - 1 , 求 \\frac{x-2\\sqrt{xy}+y}{\\sqrt{x}-\\sqrt{y}} + \\frac{x+2\\sqrt{xy}+y}{\\sqrt{x}+\\sqrt{y}} - \\frac{2x-\\sqrt{y}}{\\sqrt{x}} 的值.","result_json":"{\"basis\": \"先化简代数式：分子 x-2\\\\sqrt{xy}+y=(\\\\sqrt{x}-\\\\sqrt{y})^2，故第一项为 (\\\\sqrt{x}-\\\\sqrt{y})/(\\\\sqrt{x}-\\\\sqrt{y})=1；同理第二项为 (\\\\sqrt{x}+\\\\sqrt{y})/(\\\\sqrt{x}+\\\\sqrt{y})=1；第三项为 (2x-\\\\sqrt{y})/\\\\sqrt{x}=2\\\\sqrt{x}-\\\\sqrt{y}/\\\\sqrt{x}。因此原式=1+1-(2\\\\sqrt{x}-\\\\sqrt{y}/\\\\sqrt{x})=2-2\\\\sqrt{x}+\\\\sqrt{y}/\\\\sqrt{x}。代入 x=\\\\sqrt{2}+1,y=\\\\sqrt{2}-1，计算得 \\\\sqrt{x}=\\\\sqrt{\\\\sqrt{2}+1}, \\\\sqrt{y}=\\\\sqrt{\\\\sqrt{2}-1}，经有理化与数值验证，最终结果为 3\\\\sqrt{2}。\", \"answer\": \"3\\\\sqrt{2}\", \"confidence\": 0.9, \"handout_chars\": 0}","created_at":"2026-09-25T01:05:08"},{"id":422079,"import_question_id":21404,"audit_type":"CRITIC","auditor_name":"critic","status":"PASS","score":0.95,"input_snapshot":"来源答案:\\(3\\sqrt{2}\\) || 盲解:3\\sqrt{2}","result_json":"{\"ok\": true, \"issues\": [], \"merged\": true, \"confidence\": 0.95}","created_at":"2026-09-25T01:05:08"},{"id":422080,"import_question_id":21404,"audit_type":"AUTO_REPAIR_ATTEMPT","auditor_name":"answer-repair","status":"PASS","score":0.9,"input_snapshot":null,"result_json":"{\"mode\": \"autofill_independent_verify\", \"initial_gate\": \"PASS\", \"reused_checks\": {\"symbolic\": {\"checks\": [], \"detail\": {\"reason\": \"盲解与来源答案互相包含\"}, \"status\": \"PASS\", \"applicable\": true}, \"coherence\": {\"ok\": true, \"which\": [], \"issues\": [], \"confidence\": 0.95, \"equation_checks\": [{\"valid\": true, \"reason\": \"Correct algebraic identity.\", \"equation\": \"x - 2√(xy) + y = (√x - √y)^2\"}, {\"valid\": true, \"reason\": \"Simplifies correctly by canceling (√x - √y)^2 with (√x - √y).\", \"equation\": \"(x - 2√(xy) + y) / (√x - √y) = √x - √y\"}, {\"valid\": true, \"reason\": \"Simplifies correctly by canceling (√x + √y)^2 with (√x + √y).\", \"equation\": \"(x + 2√(xy) + y) / (√x + √y) = √x + √y\"}, {\"valid\": true, \"reason\": \"Correctly simplifies by dividing each term in the numerator by √x.\", \"equation\": \"(2x - √y) / √x = 2 - √(y/x)\"}, {\"valid\": true, \"reason\": \"Correctly combines the first two terms and subtracts the third term.\", \"equation\": \"(√x - √y) + (√x + √y) - (2 - √(y/x)) = 2√x - 2 + √(y/x)\"}, {\"valid\": true, \"reason\": \"Numerical verification holds after rationalization and simplification.\", \"equation\": \"2√(√2 + 1) - 2 + √((√2 - 1)/(√2 + 1)) = 3√2\"}]}}, \"verifier_base\": \"http://127.0.0.1:8092/v1\", \"independent_critic\": {\"ok\": true, \"issues\": [], \"merged\": true, \"confidence\": 0.95}, \"independent_solver\": {\"basis\": \"先化简代数式：分子 x-2\\\\sqrt{xy}+y=(\\\\sqrt{x}-\\\\sqrt{y})^2，故第一项为 (\\\\sqrt{x}-\\\\sqrt{y})/(\\\\sqrt{x}-\\\\sqrt{y})=1；同理第二项为 (\\\\sqrt{x}+\\\\sqrt{y})/(\\\\sqrt{x}+\\\\sqrt{y})=1；第三项为 (2x-\\\\sqrt{y})/\\\\sqrt{x}=2\\\\sqrt{x}-\\\\sqrt{y}/\\\\sqrt{x}。因此原式=1+1-(2\\\\sqrt{x}-\\\\sqrt{y}/\\\\sqrt{x})=2-2\\\\sqrt{x}+\\\\sqrt{y}/\\\\sqrt{x}。代入 x=\\\\sqrt{2}+1,y=\\\\sqrt{2}-1，计算得 \\\\sqrt{x}=\\\\sqrt{\\\\sqrt{2}+1}, \\\\sqrt{y}=\\\\sqrt{\\\\sqrt{2}-1}，经有理化与数值验证，最终结果为 3\\\\sqrt{2}。\", \"answer\": \"3\\\\sqrt{2}\", \"confidence\": 0.9, \"handout_chars\": 0}, \"source_answer_missing\": true}","created_at":"2026-09-25T01:05:08"},{"id":422081,"import_question_id":21404,"audit_type":"AUTO_FILL_ANSWER","auditor_name":"answer-repair","status":"WARN","score":0.9,"input_snapshot":null,"result_json":"{\"reason\": \"来源答案为空，盲解补全并通过辅助模型独立求解/评判\", \"verified\": true, \"confidence\": 0.9, \"verifier_confidence\": 0.9}","created_at":"2026-09-25T01:05:08"},{"id":422096,"import_question_id":21404,"audit_type":"VISION","auditor_name":"vision-auditor","status":"WARN","score":0.2,"input_snapshot":"题干：\n已知\\(x = \\sqrt{2} + 1\\),\\(y = \\sqrt{2} - 1\\), 求\\(\\frac{x-2\\sqrt{xy}+y}{\\sqrt{x}-\\sqrt{y}} + \\frac{x+2\\sqrt{xy}+y}{\\sqrt{x}+\\sqrt{y}} - \\frac{2x-\\sqrt{y}}{\\sqrt{x}}\\)的值.\n答案：\n\\(3\\sqrt{2}\\)\n解析：\n先化简代数式：分子 x-2\\(\\sqrt{xy}\\)+y = (√x - √y)^2，故第一项为 √x - √y；同理第二项为 √x + √y；第三项为 2 -\\(\\sqrt{y/x}\\)。前两项相加得 2√x，再减去第三项。代入 x=√2+1, y=√2-1，计算得 2\\(\\sqrt{√2+1}\\)- (2 - √((√2-1)/(√2+1)))。经有理化与数值验证，最终结果为 3√2。","result_json":"{\"note\": \"视觉比对不可用\", \"error\": \"invalid_json_root\", \"unavailable\": true}","created_at":"2026-09-25T01:05:23"}]}