{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0001","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5588c32f865afb419c263e19fd106c4450efafbb7a34e4f07241e5948263868","family_id":"qf-quadratic-form-validity","seed":202610034001,"semantic_template":"qf-quadratic-form-validity/{\"a\":-2,\"b\":-12,\"c\":-3,\"domain\":\"reals\",\"operation\":\"quadratic_form\"}"},"fields":{"title":"Validity of a Quadratic-Equation Form","stem":"Over the reals, is (-2)x²+(-12)x+(-3)=0 a quadratic equation in one variable? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"The quadratic coefficient is a=-2, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written."},"skills":["Validity of a Quadratic-Equation Form"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0002","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac25727e63334a669656a2932f54e411828b7a4fe394c14f6682fb60803ad88a","family_id":"qf-quadratic-form-validity","seed":202610034002,"semantic_template":"qf-quadratic-form-validity/{\"a\":-1,\"b\":-11,\"c\":-2,\"domain\":\"reals\",\"operation\":\"quadratic_form\"}"},"fields":{"title":"Validity of a Quadratic-Equation Form","stem":"Over the reals, is (-1)x²+(-11)x+(-2)=0 a quadratic equation in one variable? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"The quadratic coefficient is a=-1, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written."},"skills":["Validity of a Quadratic-Equation Form"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0003","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a422f2e283daecfa3f1901e072c0d274524e1d16ad25c07f6f94b3f36d1ba3f9","family_id":"qf-quadratic-form-validity","seed":202610034003,"semantic_template":"qf-quadratic-form-validity/{\"a\":0,\"b\":-10,\"c\":-1,\"domain\":\"reals\",\"operation\":\"quadratic_form\"}"},"fields":{"title":"Validity of a Quadratic-Equation Form","stem":"Over the reals, is (0)x²+(-10)x+(-1)=0 a quadratic equation in one variable? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"The quadratic coefficient is a=0, so the answer is No. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written."},"skills":["Validity of a Quadratic-Equation Form"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0004","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16eab3470cff5abf6300020f2edc7aa658b6475031b6be5e22dedb2f2bc7647f","family_id":"qf-quadratic-form-validity","seed":202610034004,"semantic_template":"qf-quadratic-form-validity/{\"a\":1,\"b\":-9,\"c\":0,\"domain\":\"reals\",\"operation\":\"quadratic_form\"}"},"fields":{"title":"Validity of a Quadratic-Equation Form","stem":"Over the reals, is (1)x²+(-9)x+(0)=0 a quadratic equation in one variable? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"The quadratic coefficient is a=1, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written."},"skills":["Validity of a Quadratic-Equation Form"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0005","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49b79b0302e05201e4fc77af98aaa3ef6d1b43374b36edd6daf2116be8580636","family_id":"qf-quadratic-form-validity","seed":202610034005,"semantic_template":"qf-quadratic-form-validity/{\"a\":2,\"b\":-8,\"c\":1,\"domain\":\"reals\",\"operation\":\"quadratic_form\"}"},"fields":{"title":"Validity of a Quadratic-Equation Form","stem":"Over the reals, is (2)x²+(-8)x+(1)=0 a quadratic equation in one variable? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"The quadratic coefficient is a=2, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written."},"skills":["Validity of a Quadratic-Equation Form"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0006","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cabc5b73cb85d9a2b0954a2dc27250179cf38a702053ec3b287f1e3d15671998","family_id":"qf-quadratic-form-validity","seed":202610034006,"semantic_template":"qf-quadratic-form-validity/{\"a\":-2,\"b\":-7,\"c\":2,\"domain\":\"reals\",\"operation\":\"quadratic_form\"}"},"fields":{"title":"Validity of a Quadratic-Equation Form","stem":"Over the reals, is (-2)x²+(-7)x+(2)=0 a quadratic equation in one variable? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"The quadratic coefficient is a=-2, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written."},"skills":["Validity of a Quadratic-Equation Form"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0007","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f999f3ab811d1768353379028962442cc9f97ae42eb9d51bfe7b0cc425c3091c","family_id":"qf-quadratic-form-validity","seed":202610034007,"semantic_template":"qf-quadratic-form-validity/{\"a\":-1,\"b\":-6,\"c\":3,\"domain\":\"reals\",\"operation\":\"quadratic_form\"}"},"fields":{"title":"Validity of a Quadratic-Equation Form","stem":"Over the reals, is (-1)x²+(-6)x+(3)=0 a quadratic equation in one variable? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"The quadratic coefficient is a=-1, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written."},"skills":["Validity of a Quadratic-Equation Form"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0008","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50e1c5facad31ad304bc916cd2b0a2b10cc69331a93a454add959ec3b2ed36db","family_id":"qf-quadratic-form-validity","seed":202610034008,"semantic_template":"qf-quadratic-form-validity/{\"a\":0,\"b\":-5,\"c\":-3,\"domain\":\"reals\",\"operation\":\"quadratic_form\"}"},"fields":{"title":"Validity of a Quadratic-Equation Form","stem":"Over the reals, is (0)x²+(-5)x+(-3)=0 a quadratic equation in one variable? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"The quadratic coefficient is a=0, so the answer is No. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written."},"skills":["Validity of a Quadratic-Equation Form"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0009","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bda3b5da0f4057dfea1aa37f84196ecea556ff398738719f48709cca50a9f842","family_id":"qf-quadratic-form-validity","seed":202610034009,"semantic_template":"qf-quadratic-form-validity/{\"a\":1,\"b\":-4,\"c\":-2,\"domain\":\"reals\",\"operation\":\"quadratic_form\"}"},"fields":{"title":"Validity of a Quadratic-Equation Form","stem":"Over the reals, is (1)x²+(-4)x+(-2)=0 a quadratic equation in one variable? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"The quadratic coefficient is a=1, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written."},"skills":["Validity of a Quadratic-Equation Form"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0010","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4abd0222227f283b5fabcef68ac6845c6e189b727fe3512cad6e330224670179","family_id":"qf-quadratic-form-validity","seed":202610034010,"semantic_template":"qf-quadratic-form-validity/{\"a\":2,\"b\":-3,\"c\":-1,\"domain\":\"reals\",\"operation\":\"quadratic_form\"}"},"fields":{"title":"Validity of a Quadratic-Equation Form","stem":"Over the reals, is (2)x²+(-3)x+(-1)=0 a quadratic equation in one variable? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"The quadratic coefficient is a=2, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written."},"skills":["Validity of a Quadratic-Equation Form"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0011","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b82f734e88c5bd7b028a355b583c5df6fb600317bfc702a54181147a75575a34","family_id":"qf-quadratic-form-validity","seed":202610034011,"semantic_template":"qf-quadratic-form-validity/{\"a\":-2,\"b\":-2,\"c\":0,\"domain\":\"reals\",\"operation\":\"quadratic_form\"}"},"fields":{"title":"Validity of a Quadratic-Equation Form","stem":"Over the reals, is (-2)x²+(-2)x+(0)=0 a quadratic equation in one variable? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"The quadratic coefficient is a=-2, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written."},"skills":["Validity of a Quadratic-Equation Form"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0012","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e02a5a0831bec11a9e6a9d6ed07a127bce50549c703aca7fb3436a336a54ac6e","family_id":"qf-quadratic-form-validity","seed":202610034012,"semantic_template":"qf-quadratic-form-validity/{\"a\":-1,\"b\":-1,\"c\":1,\"domain\":\"reals\",\"operation\":\"quadratic_form\"}"},"fields":{"title":"Validity of a Quadratic-Equation Form","stem":"Over the reals, is (-1)x²+(-1)x+(1)=0 a quadratic equation in one variable? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"The quadratic coefficient is a=-1, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written."},"skills":["Validity of a Quadratic-Equation Form"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0013","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38ecefc6f667389a6cbee7bb50bf97520a7c8d565a59d1927139e95722d7cc53","family_id":"qf-quadratic-form-validity","seed":202610034013,"semantic_template":"qf-quadratic-form-validity/{\"a\":0,\"b\":0,\"c\":2,\"domain\":\"reals\",\"operation\":\"quadratic_form\"}"},"fields":{"title":"Validity of a Quadratic-Equation Form","stem":"Over the reals, is (0)x²+(0)x+(2)=0 a quadratic equation in one variable? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"The quadratic coefficient is a=0, so the answer is No. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written."},"skills":["Validity of a Quadratic-Equation Form"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0014","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90d1706cfdbc638961932e955f17d2726f0d88df6afd15117bf622415a869c3e","family_id":"qf-quadratic-form-validity","seed":202610034014,"semantic_template":"qf-quadratic-form-validity/{\"a\":1,\"b\":1,\"c\":3,\"domain\":\"reals\",\"operation\":\"quadratic_form\"}"},"fields":{"title":"Validity of a Quadratic-Equation Form","stem":"Over the reals, is (1)x²+(1)x+(3)=0 a quadratic equation in one variable? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"The quadratic coefficient is a=1, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written."},"skills":["Validity of a Quadratic-Equation Form"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0015","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5fb3210a2f9d805b5e8444ea27a4d8a603548753eb110a958ff1285a60502625","family_id":"qf-quadratic-form-validity","seed":202610034015,"semantic_template":"qf-quadratic-form-validity/{\"a\":2,\"b\":2,\"c\":-3,\"domain\":\"reals\",\"operation\":\"quadratic_form\"}"},"fields":{"title":"Validity of a Quadratic-Equation Form","stem":"Over the reals, is (2)x²+(2)x+(-3)=0 a quadratic equation in one variable? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"The quadratic coefficient is a=2, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written."},"skills":["Validity of a Quadratic-Equation Form"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0016","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb57e1391cfab0eecfdcde825939a9c78f4d9b61523c200b7ea99f32b37b4a67","family_id":"qf-quadratic-form-validity","seed":202610034016,"semantic_template":"qf-quadratic-form-validity/{\"a\":-2,\"b\":3,\"c\":-2,\"domain\":\"reals\",\"operation\":\"quadratic_form\"}"},"fields":{"title":"Validity of a Quadratic-Equation Form","stem":"Over the reals, is (-2)x²+(3)x+(-2)=0 a quadratic equation in one variable? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"The quadratic coefficient is a=-2, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written."},"skills":["Validity of a Quadratic-Equation Form"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0017","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0fddbdf6286a91004237e5b1f1f613340b03dddd2da8d63fe739401848dcaa29","family_id":"qf-quadratic-form-validity","seed":202610034017,"semantic_template":"qf-quadratic-form-validity/{\"a\":-1,\"b\":4,\"c\":-1,\"domain\":\"reals\",\"operation\":\"quadratic_form\"}"},"fields":{"title":"Validity of a Quadratic-Equation Form","stem":"Over the reals, is (-1)x²+(4)x+(-1)=0 a quadratic equation in one variable? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"The quadratic coefficient is a=-1, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written."},"skills":["Validity of a Quadratic-Equation Form"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0018","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd0af92bdb9f642ffdd3441803d8dd2697f87b6bab79c3a8a1045b45de730fa1","family_id":"qf-quadratic-form-validity","seed":202610034018,"semantic_template":"qf-quadratic-form-validity/{\"a\":0,\"b\":5,\"c\":0,\"domain\":\"reals\",\"operation\":\"quadratic_form\"}"},"fields":{"title":"Validity of a Quadratic-Equation Form","stem":"Over the reals, is (0)x²+(5)x+(0)=0 a quadratic equation in one variable? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"The quadratic coefficient is a=0, so the answer is No. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written."},"skills":["Validity of a Quadratic-Equation Form"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0019","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2bf4cb5b0d3be8907fddbd542c1f4891f75828503a7bc86570b614a27249deb","family_id":"qf-quadratic-form-validity","seed":202610034019,"semantic_template":"qf-quadratic-form-validity/{\"a\":1,\"b\":6,\"c\":1,\"domain\":\"reals\",\"operation\":\"quadratic_form\"}"},"fields":{"title":"Validity of a Quadratic-Equation Form","stem":"Over the reals, is (1)x²+(6)x+(1)=0 a quadratic equation in one variable? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"The quadratic coefficient is a=1, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written."},"skills":["Validity of a Quadratic-Equation Form"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0020","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72ff3db1d978cd836fc8b64c86664870c6ac284ac5187711ba0046bfa836cdf6","family_id":"qf-quadratic-form-validity","seed":202610034020,"semantic_template":"qf-quadratic-form-validity/{\"a\":2,\"b\":7,\"c\":2,\"domain\":\"reals\",\"operation\":\"quadratic_form\"}"},"fields":{"title":"Validity of a Quadratic-Equation Form","stem":"Over the reals, is (2)x²+(7)x+(2)=0 a quadratic equation in one variable? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"The quadratic coefficient is a=2, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written."},"skills":["Validity of a Quadratic-Equation Form"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0021","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f19339d45d1cc1f826e9a16a8dc8741a2059b7b0197df379da395cad279e50ea","family_id":"qf-quadratic-form-validity","seed":202610034021,"semantic_template":"qf-quadratic-form-validity/{\"a\":-2,\"b\":8,\"c\":3,\"domain\":\"reals\",\"operation\":\"quadratic_form\"}"},"fields":{"title":"Validity of a Quadratic-Equation Form","stem":"Over the reals, is (-2)x²+(8)x+(3)=0 a quadratic equation in one variable? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"The quadratic coefficient is a=-2, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written."},"skills":["Validity of a Quadratic-Equation Form"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0022","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"304a6e0990812570280f925b5bcef3d1703d9c10c89e81f459815a0e3d97b3c2","family_id":"qf-quadratic-form-validity","seed":202610034022,"semantic_template":"qf-quadratic-form-validity/{\"a\":-1,\"b\":9,\"c\":-3,\"domain\":\"reals\",\"operation\":\"quadratic_form\"}"},"fields":{"title":"Validity of a Quadratic-Equation Form","stem":"Over the reals, is (-1)x²+(9)x+(-3)=0 a quadratic equation in one variable? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"The quadratic coefficient is a=-1, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written."},"skills":["Validity of a Quadratic-Equation Form"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0023","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afea9190a61045225efdeca4d8e3b9002d43415792f3e518fe76a209af8ef40a","family_id":"qf-quadratic-form-validity","seed":202610034023,"semantic_template":"qf-quadratic-form-validity/{\"a\":0,\"b\":10,\"c\":-2,\"domain\":\"reals\",\"operation\":\"quadratic_form\"}"},"fields":{"title":"Validity of a Quadratic-Equation Form","stem":"Over the reals, is (0)x²+(10)x+(-2)=0 a quadratic equation in one variable? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"The quadratic coefficient is a=0, so the answer is No. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written."},"skills":["Validity of a Quadratic-Equation Form"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0024","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"990fff3946c421e782f1abe4e2d464f8b88de5ee470dcc95978c956659632a27","family_id":"qf-quadratic-form-validity","seed":202610034024,"semantic_template":"qf-quadratic-form-validity/{\"a\":1,\"b\":11,\"c\":-1,\"domain\":\"reals\",\"operation\":\"quadratic_form\"}"},"fields":{"title":"Validity of a Quadratic-Equation Form","stem":"Over the reals, is (1)x²+(11)x+(-1)=0 a quadratic equation in one variable? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"The quadratic coefficient is a=1, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written."},"skills":["Validity of a Quadratic-Equation Form"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0025","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7fc0dfb26e038d7fce573688c156a4fccf437469bed59fb6535bfd9c9500562","family_id":"qf-quadratic-form-validity","seed":202610034025,"semantic_template":"qf-quadratic-form-validity/{\"a\":2,\"b\":12,\"c\":0,\"domain\":\"reals\",\"operation\":\"quadratic_form\"}"},"fields":{"title":"Validity of a Quadratic-Equation Form","stem":"Over the reals, is (2)x²+(12)x+(0)=0 a quadratic equation in one variable? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"The quadratic coefficient is a=2, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written."},"skills":["Validity of a Quadratic-Equation Form"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0026","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a0b995ed388ee4a8f47d10cd0e4d8f7ca51b196db7ceeab13f52b6484359945","family_id":"qf-quadratic-discriminant","seed":202610034026,"semantic_template":"qf-quadratic-discriminant/{\"a\":1,\"b\":0,\"c\":1,\"domain\":\"reals\",\"operation\":\"discriminant\"}"},"fields":{"title":"Discriminant and Distinct Real Roots","stem":"Over the reals, (1)x²+(0)x+(1)=0 has a≠0. Find the discriminant D and the number of distinct real roots.","answer":{"kind":"short_text","value":"D=-4; distinct real roots=0","explanation":"D=-4; distinct real roots=0. 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0027","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ed34afea4008727e91bdb829e861abae2b12c31c07ab192b873d0c9b115c534","family_id":"qf-quadratic-discriminant","seed":202610034027,"semantic_template":"qf-quadratic-discriminant/{\"a\":1,\"b\":0,\"c\":0,\"domain\":\"reals\",\"operation\":\"discriminant\"}"},"fields":{"title":"Discriminant and Distinct Real Roots","stem":"Over the reals, (1)x²+(0)x+(0)=0 has a≠0. Find the discriminant D and the number of distinct real roots.","answer":{"kind":"short_text","value":"D=0; distinct real roots=1","explanation":"D=0; distinct real roots=1. 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0028","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"318339d8f0e36dd1da465eb1d0bf0ad4f8599806d12995c01c74bb5bd25aa76b","family_id":"qf-quadratic-discriminant","seed":202610034028,"semantic_template":"qf-quadratic-discriminant/{\"a\":-1,\"b\":0,\"c\":1,\"domain\":\"reals\",\"operation\":\"discriminant\"}"},"fields":{"title":"Discriminant and Distinct Real Roots","stem":"Over the reals, (-1)x²+(0)x+(1)=0 has a≠0. Find the discriminant D and the number of distinct real roots.","answer":{"kind":"short_text","value":"D=4; distinct real roots=2","explanation":"D=4; 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0029","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"085c5c1217921c6042125a897fadbee826c948db1271396aa05abadb43b970ac","family_id":"qf-quadratic-discriminant","seed":202610034029,"semantic_template":"qf-quadratic-discriminant/{\"a\":1,\"b\":2,\"c\":2,\"domain\":\"reals\",\"operation\":\"discriminant\"}"},"fields":{"title":"Discriminant and Distinct Real Roots","stem":"Over the reals, (1)x²+(2)x+(2)=0 has a≠0. Find the discriminant D and the number of distinct real roots.","answer":{"kind":"short_text","value":"D=-4; distinct real roots=0","explanation":"D=-4; distinct real roots=0. 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0030","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2acff61b0c7ac46e164666df1bf8fb9f07eb7701bc589289fd2ba1d98b3a5c93","family_id":"qf-quadratic-discriminant","seed":202610034030,"semantic_template":"qf-quadratic-discriminant/{\"a\":1,\"b\":2,\"c\":1,\"domain\":\"reals\",\"operation\":\"discriminant\"}"},"fields":{"title":"Discriminant and Distinct Real Roots","stem":"Over the reals, (1)x²+(2)x+(1)=0 has a≠0. Find the discriminant D and the number of distinct real roots.","answer":{"kind":"short_text","value":"D=0; distinct real roots=1","explanation":"D=0; distinct real roots=1. 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0031","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4dd1a715821b3ff84aa1bc5c7a0de361bb53bceaa8be9b09772c1b0e85fb8d7d","family_id":"qf-quadratic-discriminant","seed":202610034031,"semantic_template":"qf-quadratic-discriminant/{\"a\":-1,\"b\":1,\"c\":2,\"domain\":\"reals\",\"operation\":\"discriminant\"}"},"fields":{"title":"Discriminant and Distinct Real Roots","stem":"Over the reals, (-1)x²+(1)x+(2)=0 has a≠0. Find the discriminant D and the number of distinct real roots.","answer":{"kind":"short_text","value":"D=9; distinct real roots=2","explanation":"D=9; 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0032","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"544d0bc7334d0a3ed67214b3388af97a406f02c538c9651d4c1f6ad8dc5fad3a","family_id":"qf-quadratic-discriminant","seed":202610034032,"semantic_template":"qf-quadratic-discriminant/{\"a\":1,\"b\":4,\"c\":5,\"domain\":\"reals\",\"operation\":\"discriminant\"}"},"fields":{"title":"Discriminant and Distinct Real Roots","stem":"Over the reals, (1)x²+(4)x+(5)=0 has a≠0. Find the discriminant D and the number of distinct real roots.","answer":{"kind":"short_text","value":"D=-4; distinct real roots=0","explanation":"D=-4; distinct real roots=0. 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0033","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9857dac1daccba2d3eba6e4f57c147c8ea22bb6a54ba0736a6417ce441926b90","family_id":"qf-quadratic-discriminant","seed":202610034033,"semantic_template":"qf-quadratic-discriminant/{\"a\":1,\"b\":4,\"c\":4,\"domain\":\"reals\",\"operation\":\"discriminant\"}"},"fields":{"title":"Discriminant and Distinct Real Roots","stem":"Over the reals, (1)x²+(4)x+(4)=0 has a≠0. Find the discriminant D and the number of distinct real roots.","answer":{"kind":"short_text","value":"D=0; distinct real roots=1","explanation":"D=0; distinct real roots=1. 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0034","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4aeadc2ad84c453511aee83d9210ec614ee87d300aea29438e1f8df4acb0f3e3","family_id":"qf-quadratic-discriminant","seed":202610034034,"semantic_template":"qf-quadratic-discriminant/{\"a\":-1,\"b\":2,\"c\":3,\"domain\":\"reals\",\"operation\":\"discriminant\"}"},"fields":{"title":"Discriminant and Distinct Real Roots","stem":"Over the reals, (-1)x²+(2)x+(3)=0 has a≠0. Find the discriminant D and the number of distinct real roots.","answer":{"kind":"short_text","value":"D=16; distinct real roots=2","explanation":"D=16; 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0035","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31d8eb980c68c8fbc10038dab82910a5263cb66e8d5a53e5079dbcef32a2bbfb","family_id":"qf-quadratic-discriminant","seed":202610034035,"semantic_template":"qf-quadratic-discriminant/{\"a\":1,\"b\":6,\"c\":10,\"domain\":\"reals\",\"operation\":\"discriminant\"}"},"fields":{"title":"Discriminant and Distinct Real Roots","stem":"Over the reals, (1)x²+(6)x+(10)=0 has a≠0. Find the discriminant D and the number of distinct real roots.","answer":{"kind":"short_text","value":"D=-4; distinct real roots=0","explanation":"D=-4; distinct real roots=0. 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0036","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ffdedb3d937fd9339a716d1a49e0e5acf93f95ec01820d8b0de7177e68a951df","family_id":"qf-quadratic-discriminant","seed":202610034036,"semantic_template":"qf-quadratic-discriminant/{\"a\":1,\"b\":6,\"c\":9,\"domain\":\"reals\",\"operation\":\"discriminant\"}"},"fields":{"title":"Discriminant and Distinct Real Roots","stem":"Over the reals, (1)x²+(6)x+(9)=0 has a≠0. Find the discriminant D and the number of distinct real roots.","answer":{"kind":"short_text","value":"D=0; distinct real roots=1","explanation":"D=0; distinct real roots=1. 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0037","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d435e2fd0e42efa10b09be54d1d4e7495a97013544403fdb776d601b469d9e2","family_id":"qf-quadratic-discriminant","seed":202610034037,"semantic_template":"qf-quadratic-discriminant/{\"a\":-1,\"b\":3,\"c\":4,\"domain\":\"reals\",\"operation\":\"discriminant\"}"},"fields":{"title":"Discriminant and Distinct Real Roots","stem":"Over the reals, (-1)x²+(3)x+(4)=0 has a≠0. Find the discriminant D and the number of distinct real roots.","answer":{"kind":"short_text","value":"D=25; distinct real roots=2","explanation":"D=25; 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0038","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b81f97162210eaea6b77a5fde05268d81e5439ac256cdc8134bfd44f7efce8ea","family_id":"qf-quadratic-discriminant","seed":202610034038,"semantic_template":"qf-quadratic-discriminant/{\"a\":1,\"b\":8,\"c\":17,\"domain\":\"reals\",\"operation\":\"discriminant\"}"},"fields":{"title":"Discriminant and Distinct Real Roots","stem":"Over the reals, (1)x²+(8)x+(17)=0 has a≠0. Find the discriminant D and the number of distinct real roots.","answer":{"kind":"short_text","value":"D=-4; distinct real roots=0","explanation":"D=-4; distinct real roots=0. 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0039","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22d57a7e5e4397ec692d9acca0409eda14969adfdbc31319b2591cccb4a2b3ff","family_id":"qf-quadratic-discriminant","seed":202610034039,"semantic_template":"qf-quadratic-discriminant/{\"a\":1,\"b\":8,\"c\":16,\"domain\":\"reals\",\"operation\":\"discriminant\"}"},"fields":{"title":"Discriminant and Distinct Real Roots","stem":"Over the reals, (1)x²+(8)x+(16)=0 has a≠0. Find the discriminant D and the number of distinct real roots.","answer":{"kind":"short_text","value":"D=0; distinct real roots=1","explanation":"D=0; distinct real roots=1. 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0040","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d46b0deee08eff4c4294ff9811e98d1b519aeff332ed7e1c55641171f6a882e","family_id":"qf-quadratic-discriminant","seed":202610034040,"semantic_template":"qf-quadratic-discriminant/{\"a\":-1,\"b\":4,\"c\":5,\"domain\":\"reals\",\"operation\":\"discriminant\"}"},"fields":{"title":"Discriminant and Distinct Real Roots","stem":"Over the reals, (-1)x²+(4)x+(5)=0 has a≠0. Find the discriminant D and the number of distinct real roots.","answer":{"kind":"short_text","value":"D=36; distinct real roots=2","explanation":"D=36; 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0041","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e08b2c330bf92550d3b236f090f76da202a0c7373bfb5b738c89c839486363dd","family_id":"qf-quadratic-discriminant","seed":202610034041,"semantic_template":"qf-quadratic-discriminant/{\"a\":1,\"b\":10,\"c\":26,\"domain\":\"reals\",\"operation\":\"discriminant\"}"},"fields":{"title":"Discriminant and Distinct Real Roots","stem":"Over the reals, (1)x²+(10)x+(26)=0 has a≠0. Find the discriminant D and the number of distinct real roots.","answer":{"kind":"short_text","value":"D=-4; distinct real roots=0","explanation":"D=-4; distinct real roots=0. 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0042","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b6cd070f13ed63ad474bdb1c735de2abfed7b16c3ba0468816aea09da4b3c72f","family_id":"qf-quadratic-discriminant","seed":202610034042,"semantic_template":"qf-quadratic-discriminant/{\"a\":1,\"b\":10,\"c\":25,\"domain\":\"reals\",\"operation\":\"discriminant\"}"},"fields":{"title":"Discriminant and Distinct Real Roots","stem":"Over the reals, (1)x²+(10)x+(25)=0 has a≠0. Find the discriminant D and the number of distinct real roots.","answer":{"kind":"short_text","value":"D=0; distinct real roots=1","explanation":"D=0; distinct real roots=1. 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."}}
{"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0044","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d89a95e2dbcd9d9d426d9ea1c4b782d6a96ee505e0878f7e1336dff6b342e08","family_id":"qf-quadratic-discriminant","seed":202610034044,"semantic_template":"qf-quadratic-discriminant/{\"a\":1,\"b\":12,\"c\":37,\"domain\":\"reals\",\"operation\":\"discriminant\"}"},"fields":{"title":"Discriminant and Distinct Real Roots","stem":"Over the reals, (1)x²+(12)x+(37)=0 has a≠0. Find the discriminant D and the number of distinct real roots.","answer":{"kind":"short_text","value":"D=-4; distinct real roots=0","explanation":"D=-4; distinct real roots=0. 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0045","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6485bdb172dca5fc97ee5983d8ede05df7c70bdefc229c53b4154794b8e2e189","family_id":"qf-quadratic-discriminant","seed":202610034045,"semantic_template":"qf-quadratic-discriminant/{\"a\":1,\"b\":12,\"c\":36,\"domain\":\"reals\",\"operation\":\"discriminant\"}"},"fields":{"title":"Discriminant and Distinct Real Roots","stem":"Over the reals, (1)x²+(12)x+(36)=0 has a≠0. Find the discriminant D and the number of distinct real roots.","answer":{"kind":"short_text","value":"D=0; distinct real roots=1","explanation":"D=0; distinct real roots=1. 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0046","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a21e101c02b87354d6b83ce250eb7772076cfefd913cec2f6c5cf29ece1a5f69","family_id":"qf-quadratic-discriminant","seed":202610034046,"semantic_template":"qf-quadratic-discriminant/{\"a\":-1,\"b\":6,\"c\":7,\"domain\":\"reals\",\"operation\":\"discriminant\"}"},"fields":{"title":"Discriminant and Distinct Real Roots","stem":"Over the reals, (-1)x²+(6)x+(7)=0 has a≠0. Find the discriminant D and the number of distinct real roots.","answer":{"kind":"short_text","value":"D=64; distinct real roots=2","explanation":"D=64; 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0047","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abd6c156dfbc65646f700114cfc5e418e0e6687c8a5e6c2981febf7267994b23","family_id":"qf-quadratic-discriminant","seed":202610034047,"semantic_template":"qf-quadratic-discriminant/{\"a\":1,\"b\":14,\"c\":50,\"domain\":\"reals\",\"operation\":\"discriminant\"}"},"fields":{"title":"Discriminant and Distinct Real Roots","stem":"Over the reals, (1)x²+(14)x+(50)=0 has a≠0. Find the discriminant D and the number of distinct real roots.","answer":{"kind":"short_text","value":"D=-4; distinct real roots=0","explanation":"D=-4; distinct real roots=0. 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0048","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e84fce04ad1a937de20b08b8f77f8257f1fcb3aea01afba32e6b3824dbcd21dc","family_id":"qf-quadratic-discriminant","seed":202610034048,"semantic_template":"qf-quadratic-discriminant/{\"a\":1,\"b\":14,\"c\":49,\"domain\":\"reals\",\"operation\":\"discriminant\"}"},"fields":{"title":"Discriminant and Distinct Real Roots","stem":"Over the reals, (1)x²+(14)x+(49)=0 has a≠0. Find the discriminant D and the number of distinct real roots.","answer":{"kind":"short_text","value":"D=0; distinct real roots=1","explanation":"D=0; distinct real roots=1. 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0049","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8073ab99a0ab97eacdf2534e7921d6985593156340dac4f9a74569220679e6bc","family_id":"qf-quadratic-discriminant","seed":202610034049,"semantic_template":"qf-quadratic-discriminant/{\"a\":-1,\"b\":7,\"c\":8,\"domain\":\"reals\",\"operation\":\"discriminant\"}"},"fields":{"title":"Discriminant and Distinct Real Roots","stem":"Over the reals, (-1)x²+(7)x+(8)=0 has a≠0. Find the discriminant D and the number of distinct real roots.","answer":{"kind":"short_text","value":"D=81; distinct real roots=2","explanation":"D=81; 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0050","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4e744401723c2fac1c8d57d7cfbe377c52d1394ba25594384e2de45357c6901","family_id":"qf-quadratic-discriminant","seed":202610034050,"semantic_template":"qf-quadratic-discriminant/{\"a\":1,\"b\":16,\"c\":65,\"domain\":\"reals\",\"operation\":\"discriminant\"}"},"fields":{"title":"Discriminant and Distinct Real Roots","stem":"Over the reals, (1)x²+(16)x+(65)=0 has a≠0. Find the discriminant D and the number of distinct real roots.","answer":{"kind":"short_text","value":"D=-4; distinct real roots=0","explanation":"D=-4; distinct real roots=0. 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0051","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc9f769908229ec1502a5f11deab7aa6a8cacb997a3982924b07216e238c3a35","family_id":"qf-quadratic-root-verification","seed":202610034051,"semantic_template":"qf-quadratic-root-verification/{\"a\":2,\"b\":16,\"c\":24,\"domain\":\"reals\",\"operation\":\"root_verification\",\"p\":\"-6\",\"q\":\"1\"}"},"fields":{"title":"Exact Substitution of a Rational Root Candidate","stem":"Over the reals, (2)x²+(16)x+(24)=0 has a≠0. Is the reduced rational candidate x=-6/1, with positive denominator, a solution?","answer":{"kind":"short_text","value":"Yes","explanation":"Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 0; comparison with zero gives Yes."},"skills":["Exact Substitution of a Rational Root Candidate"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0052","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a28edc0113817e08bdcba5d59e6695e5440c6e35e59a55f232125d0ba54430b1","family_id":"qf-quadratic-root-verification","seed":202610034052,"semantic_template":"qf-quadratic-root-verification/{\"a\":-4,\"b\":-26,\"c\":-22,\"domain\":\"reals\",\"operation\":\"root_verification\",\"p\":\"-5\",\"q\":\"1\"}"},"fields":{"title":"Exact Substitution of a Rational Root Candidate","stem":"Over the reals, (-4)x²+(-26)x+(-22)=0 has a≠0. Is the reduced rational candidate x=-5/1, with positive denominator, a solution?","answer":{"kind":"short_text","value":"No","explanation":"Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 8; comparison with zero gives No."},"skills":["Exact Substitution of a Rational Root Candidate"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0053","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f782579af958d734742c9c403a73d5046e67b183f48d719facf5b1aec765f8ca","family_id":"qf-quadratic-root-verification","seed":202610034053,"semantic_template":"qf-quadratic-root-verification/{\"a\":6,\"b\":30,\"c\":0,\"domain\":\"reals\",\"operation\":\"root_verification\",\"p\":\"-5\",\"q\":\"1\"}"},"fields":{"title":"Exact Substitution of a Rational Root Candidate","stem":"Over the reals, (6)x²+(30)x+(0)=0 has a≠0. Is the reduced rational candidate x=-5/1, with positive denominator, a solution?","answer":{"kind":"short_text","value":"Yes","explanation":"Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 0; comparison with zero gives Yes."},"skills":["Exact Substitution of a Rational Root Candidate"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0054","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5e3a3b76ed6ddc1d65e077e2c031e51607b6a577987cecda0506bb7491bda50","family_id":"qf-quadratic-root-verification","seed":202610034054,"semantic_template":"qf-quadratic-root-verification/{\"a\":-2,\"b\":-7,\"c\":9,\"domain\":\"reals\",\"operation\":\"root_verification\",\"p\":\"-4\",\"q\":\"1\"}"},"fields":{"title":"Exact Substitution of a Rational Root Candidate","stem":"Over the reals, (-2)x²+(-7)x+(9)=0 has a≠0. Is the reduced rational candidate x=-4/1, with positive denominator, a solution?","answer":{"kind":"short_text","value":"No","explanation":"Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 5; comparison with zero gives No."},"skills":["Exact Substitution of a Rational Root Candidate"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0055","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92a09a00a3719f236fadec08741184c1eaf99684dbe1414b6ac4f2b4c89578c6","family_id":"qf-quadratic-root-verification","seed":202610034055,"semantic_template":"qf-quadratic-root-verification/{\"a\":4,\"b\":8,\"c\":-32,\"domain\":\"reals\",\"operation\":\"root_verification\",\"p\":\"-4\",\"q\":\"1\"}"},"fields":{"title":"Exact Substitution of a Rational Root Candidate","stem":"Over the reals, (4)x²+(8)x+(-32)=0 has a≠0. Is the reduced rational candidate x=-4/1, with positive denominator, a solution?","answer":{"kind":"short_text","value":"Yes","explanation":"Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 0; comparison with zero gives Yes."},"skills":["Exact Substitution of a Rational Root Candidate"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0056","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c00e09618d061ca22339f28de516b347cc6dc21ba1660837472d4f7a420f04fc","family_id":"qf-quadratic-root-verification","seed":202610034056,"semantic_template":"qf-quadratic-root-verification/{\"a\":-6,\"b\":-33,\"c\":-42,\"domain\":\"reals\",\"operation\":\"root_verification\",\"p\":\"-3\",\"q\":\"1\"}"},"fields":{"title":"Exact Substitution of a Rational Root Candidate","stem":"Over the reals, (-6)x²+(-33)x+(-42)=0 has a≠0. Is the reduced rational candidate x=-3/1, with positive denominator, a solution?","answer":{"kind":"short_text","value":"No","explanation":"Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 3; comparison with zero gives No."},"skills":["Exact Substitution of a Rational Root Candidate"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0057","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74360546509e077681118cc9f312a833e19fab4be6c42123f2d92ef2a5590d4d","family_id":"qf-quadratic-root-verification","seed":202610034057,"semantic_template":"qf-quadratic-root-verification/{\"a\":2,\"b\":8,\"c\":6,\"domain\":\"reals\",\"operation\":\"root_verification\",\"p\":\"-3\",\"q\":\"1\"}"},"fields":{"title":"Exact Substitution of a Rational Root Candidate","stem":"Over the reals, (2)x²+(8)x+(6)=0 has a≠0. Is the reduced rational candidate x=-3/1, with positive denominator, a solution?","answer":{"kind":"short_text","value":"Yes","explanation":"Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 0; comparison with zero gives Yes."},"skills":["Exact Substitution of a Rational Root Candidate"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0058","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51931a19463d9cac66f29fe17490cae290ece3dec052f91ee135236c423220ee","family_id":"qf-quadratic-root-verification","seed":202610034058,"semantic_template":"qf-quadratic-root-verification/{\"a\":-4,\"b\":-10,\"c\":0,\"domain\":\"reals\",\"operation\":\"root_verification\",\"p\":\"-2\",\"q\":\"1\"}"},"fields":{"title":"Exact Substitution of a Rational Root Candidate","stem":"Over the reals, (-4)x²+(-10)x+(0)=0 has a≠0. Is the reduced rational candidate x=-2/1, with positive denominator, a solution?","answer":{"kind":"short_text","value":"No","explanation":"Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 4; comparison with zero gives No."},"skills":["Exact Substitution of a Rational Root Candidate"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0059","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f10faafe85992a20e30d5d36ea1ae0d9007a417049df4b78965b54bec8e5c819","family_id":"qf-quadratic-root-verification","seed":202610034059,"semantic_template":"qf-quadratic-root-verification/{\"a\":6,\"b\":6,\"c\":-12,\"domain\":\"reals\",\"operation\":\"root_verification\",\"p\":\"-2\",\"q\":\"1\"}"},"fields":{"title":"Exact Substitution of a Rational Root Candidate","stem":"Over the reals, (6)x²+(6)x+(-12)=0 has a≠0. Is the reduced rational candidate x=-2/1, with positive denominator, a solution?","answer":{"kind":"short_text","value":"Yes","explanation":"Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 0; comparison with zero gives Yes."},"skills":["Exact Substitution of a Rational Root Candidate"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0060","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1035d899238eec4aa903d0a6e5e40b99b8ada262cbc0f4aea8044b9a7bd1a22","family_id":"qf-quadratic-root-verification","seed":202610034060,"semantic_template":"qf-quadratic-root-verification/{\"a\":-2,\"b\":1,\"c\":6,\"domain\":\"reals\",\"operation\":\"root_verification\",\"p\":\"-1\",\"q\":\"1\"}"},"fields":{"title":"Exact Substitution of a Rational Root Candidate","stem":"Over the reals, (-2)x²+(1)x+(6)=0 has a≠0. Is the reduced rational candidate x=-1/1, with positive denominator, a solution?","answer":{"kind":"short_text","value":"No","explanation":"Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 3; comparison with zero gives No."},"skills":["Exact Substitution of a Rational Root Candidate"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0061","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4167e549e5d056c95860e532454b8ed6e54fd02af2af6e0efe7cf894efc018ef","family_id":"qf-quadratic-root-verification","seed":202610034061,"semantic_template":"qf-quadratic-root-verification/{\"a\":4,\"b\":12,\"c\":8,\"domain\":\"reals\",\"operation\":\"root_verification\",\"p\":\"-1\",\"q\":\"1\"}"},"fields":{"title":"Exact Substitution of a Rational Root Candidate","stem":"Over the reals, (4)x²+(12)x+(8)=0 has a≠0. Is the reduced rational candidate x=-1/1, with positive denominator, a solution?","answer":{"kind":"short_text","value":"Yes","explanation":"Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 0; comparison with zero gives Yes."},"skills":["Exact Substitution of a Rational Root Candidate"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0062","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15d3d2a492e6cedfc056aca2e8e401c78824f90eff18aeb48c71f9d648fff0bc","family_id":"qf-quadratic-root-verification","seed":202610034062,"semantic_template":"qf-quadratic-root-verification/{\"a\":-6,\"b\":-9,\"c\":-3,\"domain\":\"reals\",\"operation\":\"root_verification\",\"p\":\"0\",\"q\":\"1\"}"},"fields":{"title":"Exact Substitution of a Rational Root Candidate","stem":"Over the reals, (-6)x²+(-9)x+(-3)=0 has a≠0. Is the reduced rational candidate x=0/1, with positive denominator, a solution?","answer":{"kind":"short_text","value":"No","explanation":"Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is -3; comparison with zero gives No."},"skills":["Exact Substitution of a Rational Root Candidate"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0063","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c8221899a7438c78cf40160f44d8f1d77c11ea2220da7be5ef7e3eb9ea1281ab","family_id":"qf-quadratic-root-verification","seed":202610034063,"semantic_template":"qf-quadratic-root-verification/{\"a\":2,\"b\":0,\"c\":0,\"domain\":\"reals\",\"operation\":\"root_verification\",\"p\":\"0\",\"q\":\"1\"}"},"fields":{"title":"Exact Substitution of a Rational Root Candidate","stem":"Over the reals, (2)x²+(0)x+(0)=0 has a≠0. Is the reduced rational candidate x=0/1, with positive denominator, a solution?","answer":{"kind":"short_text","value":"Yes","explanation":"Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 0; comparison with zero gives Yes."},"skills":["Exact Substitution of a Rational Root Candidate"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0064","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"970db2aa8c4319bf588af35cbda3116bb8bb04927edfaa638528469fbb28a982","family_id":"qf-quadratic-root-verification","seed":202610034064,"semantic_template":"qf-quadratic-root-verification/{\"a\":-4,\"b\":6,\"c\":-2,\"domain\":\"reals\",\"operation\":\"root_verification\",\"p\":\"1\",\"q\":\"1\"}"},"fields":{"title":"Exact Substitution of a Rational Root Candidate","stem":"Over the reals, (-4)x²+(6)x+(-2)=0 has a≠0. Is the reduced rational candidate x=1/1, with positive denominator, a solution?","answer":{"kind":"short_text","value":"Yes","explanation":"Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 0; comparison with zero gives Yes."},"skills":["Exact Substitution of a Rational Root Candidate"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0065","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e940e80d70bac44dd1689980f2f2c4071b971c84ef24185d206baa565cfaf2db","family_id":"qf-quadratic-root-verification","seed":202610034065,"semantic_template":"qf-quadratic-root-verification/{\"a\":6,\"b\":-18,\"c\":12,\"domain\":\"reals\",\"operation\":\"root_verification\",\"p\":\"1\",\"q\":\"1\"}"},"fields":{"title":"Exact Substitution of a Rational Root Candidate","stem":"Over the reals, (6)x²+(-18)x+(12)=0 has a≠0. Is the reduced rational candidate x=1/1, with positive denominator, a solution?","answer":{"kind":"short_text","value":"Yes","explanation":"Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 0; comparison with zero gives Yes."},"skills":["Exact Substitution of a Rational Root Candidate"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0066","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e417f62c702c1946871c4fc71ba8a118733ea43245c90b2e2597163f851f94e0","family_id":"qf-quadratic-root-verification","seed":202610034066,"semantic_template":"qf-quadratic-root-verification/{\"a\":-2,\"b\":-1,\"c\":6,\"domain\":\"reals\",\"operation\":\"root_verification\",\"p\":\"2\",\"q\":\"1\"}"},"fields":{"title":"Exact Substitution of a Rational Root Candidate","stem":"Over the reals, (-2)x²+(-1)x+(6)=0 has a≠0. Is the reduced rational candidate x=2/1, with positive denominator, a solution?","answer":{"kind":"short_text","value":"No","explanation":"Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is -4; comparison with zero gives No."},"skills":["Exact Substitution of a Rational Root Candidate"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0067","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac15e654ba6bd6c0a9fcf64785142a14e2cdaf388241a781e205d9e948612da1","family_id":"qf-quadratic-root-verification","seed":202610034067,"semantic_template":"qf-quadratic-root-verification/{\"a\":4,\"b\":-4,\"c\":-8,\"domain\":\"reals\",\"operation\":\"root_verification\",\"p\":\"2\",\"q\":\"1\"}"},"fields":{"title":"Exact Substitution of a Rational Root Candidate","stem":"Over the reals, (4)x²+(-4)x+(-8)=0 has a≠0. Is the reduced rational candidate x=2/1, with positive denominator, a solution?","answer":{"kind":"short_text","value":"Yes","explanation":"Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 0; comparison with zero gives Yes."},"skills":["Exact Substitution of a Rational Root Candidate"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0068","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8fa8cafe56387c63d7c69efee7c1bdeb863490cdaa7c02c21d0d632a29df555b","family_id":"qf-quadratic-root-verification","seed":202610034068,"semantic_template":"qf-quadratic-root-verification/{\"a\":-6,\"b\":15,\"c\":0,\"domain\":\"reals\",\"operation\":\"root_verification\",\"p\":\"3\",\"q\":\"1\"}"},"fields":{"title":"Exact Substitution of a Rational Root Candidate","stem":"Over the reals, (-6)x²+(15)x+(0)=0 has a≠0. Is the reduced rational candidate x=3/1, with positive denominator, a solution?","answer":{"kind":"short_text","value":"No","explanation":"Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is -9; comparison with zero gives No."},"skills":["Exact Substitution of a Rational Root Candidate"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0069","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bea3a17434a8089e8f84fe99112450074b3d5620d52dd8f5370a50c9d7e37f7e","family_id":"qf-quadratic-root-verification","seed":202610034069,"semantic_template":"qf-quadratic-root-verification/{\"a\":2,\"b\":-8,\"c\":6,\"domain\":\"reals\",\"operation\":\"root_verification\",\"p\":\"3\",\"q\":\"1\"}"},"fields":{"title":"Exact Substitution of a Rational Root Candidate","stem":"Over the reals, (2)x²+(-8)x+(6)=0 has a≠0. Is the reduced rational candidate x=3/1, with positive denominator, a solution?","answer":{"kind":"short_text","value":"Yes","explanation":"Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 0; comparison with zero gives Yes."},"skills":["Exact Substitution of a Rational Root Candidate"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0070","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac9f9233017c425ba63b728fb583e4a2c6dff0e07dd3a97558e8c1f93db8149b","family_id":"qf-quadratic-root-verification","seed":202610034070,"semantic_template":"qf-quadratic-root-verification/{\"a\":-4,\"b\":22,\"c\":-28,\"domain\":\"reals\",\"operation\":\"root_verification\",\"p\":\"4\",\"q\":\"1\"}"},"fields":{"title":"Exact Substitution of a Rational Root Candidate","stem":"Over the reals, (-4)x²+(22)x+(-28)=0 has a≠0. Is the reduced rational candidate x=4/1, with positive denominator, a solution?","answer":{"kind":"short_text","value":"No","explanation":"Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is -4; comparison with zero gives No."},"skills":["Exact Substitution of a Rational Root Candidate"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0071","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b9ccfea7cd17e0d27eb5f2014765795a59db114a19c8b0c82adfc085e2d2431","family_id":"qf-quadratic-root-verification","seed":202610034071,"semantic_template":"qf-quadratic-root-verification/{\"a\":6,\"b\":-12,\"c\":-48,\"domain\":\"reals\",\"operation\":\"root_verification\",\"p\":\"4\",\"q\":\"1\"}"},"fields":{"title":"Exact Substitution of a Rational Root Candidate","stem":"Over the reals, (6)x²+(-12)x+(-48)=0 has a≠0. Is the reduced rational candidate x=4/1, with positive denominator, a solution?","answer":{"kind":"short_text","value":"Yes","explanation":"Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 0; comparison with zero gives Yes."},"skills":["Exact Substitution of a Rational Root Candidate"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0072","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c8cf04bb4361e49a3f0dfe6ea8052e2392979d8636a54d44ea64cd0841a6c19e","family_id":"qf-quadratic-root-verification","seed":202610034072,"semantic_template":"qf-quadratic-root-verification/{\"a\":-2,\"b\":7,\"c\":9,\"domain\":\"reals\",\"operation\":\"root_verification\",\"p\":\"5\",\"q\":\"1\"}"},"fields":{"title":"Exact Substitution of a Rational Root Candidate","stem":"Over the reals, (-2)x²+(7)x+(9)=0 has a≠0. Is the reduced rational candidate x=5/1, with positive denominator, a solution?","answer":{"kind":"short_text","value":"No","explanation":"Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is -6; comparison with zero gives No."},"skills":["Exact Substitution of a Rational Root Candidate"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0073","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9981aa71e0b8410cd3ee6a0f74c22f13e536ccabbc430965e318e1c49b363402","family_id":"qf-quadratic-root-verification","seed":202610034073,"semantic_template":"qf-quadratic-root-verification/{\"a\":4,\"b\":-20,\"c\":0,\"domain\":\"reals\",\"operation\":\"root_verification\",\"p\":\"5\",\"q\":\"1\"}"},"fields":{"title":"Exact Substitution of a Rational Root Candidate","stem":"Over the reals, (4)x²+(-20)x+(0)=0 has a≠0. Is the reduced rational candidate x=5/1, with positive denominator, a solution?","answer":{"kind":"short_text","value":"Yes","explanation":"Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 0; comparison with zero gives Yes."},"skills":["Exact Substitution of a Rational Root Candidate"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0074","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf6ca3f007f43d24081059b23e032522031331696e2541a36350b97642b154e7","family_id":"qf-quadratic-root-verification","seed":202610034074,"semantic_template":"qf-quadratic-root-verification/{\"a\":-6,\"b\":39,\"c\":-33,\"domain\":\"reals\",\"operation\":\"root_verification\",\"p\":\"6\",\"q\":\"1\"}"},"fields":{"title":"Exact Substitution of a Rational Root Candidate","stem":"Over the reals, (-6)x²+(39)x+(-33)=0 has a≠0. Is the reduced rational candidate x=6/1, with positive denominator, a solution?","answer":{"kind":"short_text","value":"No","explanation":"Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is -15; comparison with zero gives No."},"skills":["Exact Substitution of a Rational Root Candidate"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0075","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b7356ebbaffbf2fdc0dcf26af4992245b383f8b2454b5a9628d254a39fbcfb4","family_id":"qf-quadratic-root-verification","seed":202610034075,"semantic_template":"qf-quadratic-root-verification/{\"a\":2,\"b\":-16,\"c\":24,\"domain\":\"reals\",\"operation\":\"root_verification\",\"p\":\"6\",\"q\":\"1\"}"},"fields":{"title":"Exact Substitution of a Rational Root Candidate","stem":"Over the reals, (2)x²+(-16)x+(24)=0 has a≠0. Is the reduced rational candidate x=6/1, with positive denominator, a solution?","answer":{"kind":"short_text","value":"Yes","explanation":"Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 0; comparison with zero gives Yes."},"skills":["Exact Substitution of a Rational Root Candidate"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0076","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6552530945369e3ec5d54d8ef713523c794618b6c7d4a15f30747d1a6c64bb69","family_id":"qf-quadratic-exact-roots","seed":202610034076,"semantic_template":"qf-quadratic-exact-roots/{\"a\":1,\"b\":0,\"c\":0,\"domain\":\"reals\",\"operation\":\"exact_roots\"}"},"fields":{"title":"Exact Root Set of a Quadratic Equation","stem":"Over the reals, (1)x²+(0)x+(0)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.","answer":{"kind":"expression","value":"{0/1}","explanation":"D=0. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {0/1}. The comparison is exact."},"skills":["Exact Root Set of a Quadratic Equation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0077","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50b07ab0cada2017f3c0b812b9b3f174e199d89f3666903af5ceb77647375087","family_id":"qf-quadratic-exact-roots","seed":202610034077,"semantic_template":"qf-quadratic-exact-roots/{\"a\":-1,\"b\":2,\"c\":-1,\"domain\":\"reals\",\"operation\":\"exact_roots\"}"},"fields":{"title":"Exact Root Set of a Quadratic Equation","stem":"Over the reals, (-1)x²+(2)x+(-1)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.","answer":{"kind":"expression","value":"{1/1}","explanation":"D=0. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {1/1}. The comparison is exact."},"skills":["Exact Root Set of a Quadratic Equation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0078","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c8924e7dfcfc185dccd41fda841cf8229c93ab6937d58ccf8041df281fe4d1b6","family_id":"qf-quadratic-exact-roots","seed":202610034078,"semantic_template":"qf-quadratic-exact-roots/{\"a\":6,\"b\":32,\"c\":10,\"domain\":\"reals\",\"operation\":\"exact_roots\"}"},"fields":{"title":"Exact Root Set of a Quadratic Equation","stem":"Over the reals, (6)x²+(32)x+(10)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.","answer":{"kind":"expression","value":"{-5/1, -1/3}","explanation":"D=784. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {-5/1, -1/3}. The comparison is exact."},"skills":["Exact Root Set of a Quadratic Equation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0079","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23408bc48c56211d155e30fe5cbd1de2298e1291cf44e6de1e8feb7f3f458a71","family_id":"qf-quadratic-exact-roots","seed":202610034079,"semantic_template":"qf-quadratic-exact-roots/{\"a\":-6,\"b\":-27,\"c\":0,\"domain\":\"reals\",\"operation\":\"exact_roots\"}"},"fields":{"title":"Exact Root Set of a Quadratic Equation","stem":"Over the reals, (-6)x²+(-27)x+(0)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.","answer":{"kind":"expression","value":"{-9/2, 0/1}","explanation":"D=729. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {-9/2, 0/1}. The comparison is exact."},"skills":["Exact Root Set of a Quadratic Equation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0080","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf7d10e7888fed3f6513b59e7722dda7ca8b4dfe363ccd8ac7b13b4ff290732d","family_id":"qf-quadratic-exact-roots","seed":202610034080,"semantic_template":"qf-quadratic-exact-roots/{\"a\":6,\"b\":22,\"c\":-8,\"domain\":\"reals\",\"operation\":\"exact_roots\"}"},"fields":{"title":"Exact Root Set of a Quadratic Equation","stem":"Over the reals, (6)x²+(22)x+(-8)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.","answer":{"kind":"expression","value":"{-4/1, 1/3}","explanation":"D=676. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {-4/1, 1/3}. The comparison is exact."},"skills":["Exact Root Set of a Quadratic Equation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0081","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2619d962c843d35fc74f24e59a74ea6b2dd1beb74c83e2754c6243d179b13f69","family_id":"qf-quadratic-exact-roots","seed":202610034081,"semantic_template":"qf-quadratic-exact-roots/{\"a\":-6,\"b\":-17,\"c\":14,\"domain\":\"reals\",\"operation\":\"exact_roots\"}"},"fields":{"title":"Exact Root Set of a Quadratic Equation","stem":"Over the reals, (-6)x²+(-17)x+(14)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.","answer":{"kind":"expression","value":"{-7/2, 2/3}","explanation":"D=625. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {-7/2, 2/3}. The comparison is exact."},"skills":["Exact Root Set of a Quadratic Equation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0082","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9daef35591e0e6332c57c163db23c6ee830a311c215be7c5e04ca69819b673b","family_id":"qf-quadratic-exact-roots","seed":202610034082,"semantic_template":"qf-quadratic-exact-roots/{\"a\":6,\"b\":12,\"c\":-18,\"domain\":\"reals\",\"operation\":\"exact_roots\"}"},"fields":{"title":"Exact Root Set of a Quadratic Equation","stem":"Over the reals, (6)x²+(12)x+(-18)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.","answer":{"kind":"expression","value":"{-3/1, 1/1}","explanation":"D=576. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {-3/1, 1/1}. The comparison is exact."},"skills":["Exact Root Set of a Quadratic Equation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0083","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4c6fd23cb9974cf9d0a8cd9726cb617901015b9e668235bc3038de5646d4561","family_id":"qf-quadratic-exact-roots","seed":202610034083,"semantic_template":"qf-quadratic-exact-roots/{\"a\":-6,\"b\":-21,\"c\":-15,\"domain\":\"reals\",\"operation\":\"exact_roots\"}"},"fields":{"title":"Exact Root Set of a Quadratic Equation","stem":"Over the reals, (-6)x²+(-21)x+(-15)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.","answer":{"kind":"expression","value":"{-5/2, -1/1}","explanation":"D=81. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {-5/2, -1/1}. The comparison is exact."},"skills":["Exact Root Set of a Quadratic Equation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0084","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6be497bc0b3c2b6649fde2e93e0daeafc4ffab3c8543ac4630515a2666a98fa3","family_id":"qf-quadratic-exact-roots","seed":202610034084,"semantic_template":"qf-quadratic-exact-roots/{\"a\":6,\"b\":16,\"c\":8,\"domain\":\"reals\",\"operation\":\"exact_roots\"}"},"fields":{"title":"Exact Root Set of a Quadratic Equation","stem":"Over the reals, (6)x²+(16)x+(8)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.","answer":{"kind":"expression","value":"{-2/1, -2/3}","explanation":"D=64. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {-2/1, -2/3}. The comparison is exact."},"skills":["Exact Root Set of a Quadratic Equation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0085","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0022e55cba3b8eb40696dfb3d4910cb71dd61b0eef8954567fe61b9d9fc0ee93","family_id":"qf-quadratic-exact-roots","seed":202610034085,"semantic_template":"qf-quadratic-exact-roots/{\"a\":-6,\"b\":-11,\"c\":-3,\"domain\":\"reals\",\"operation\":\"exact_roots\"}"},"fields":{"title":"Exact Root Set of a Quadratic Equation","stem":"Over the reals, (-6)x²+(-11)x+(-3)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.","answer":{"kind":"expression","value":"{-3/2, -1/3}","explanation":"D=49. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {-3/2, -1/3}. The comparison is exact."},"skills":["Exact Root Set of a Quadratic Equation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0086","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2781fdcf091a4ad542780200f8adc0147d984aadaf6d94b63799fd133f9fa15","family_id":"qf-quadratic-exact-roots","seed":202610034086,"semantic_template":"qf-quadratic-exact-roots/{\"a\":6,\"b\":6,\"c\":0,\"domain\":\"reals\",\"operation\":\"exact_roots\"}"},"fields":{"title":"Exact Root Set of a Quadratic Equation","stem":"Over the reals, (6)x²+(6)x+(0)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.","answer":{"kind":"expression","value":"{-1/1, 0/1}","explanation":"D=36. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {-1/1, 0/1}. The comparison is exact."},"skills":["Exact Root Set of a Quadratic Equation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0087","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"729103758e39b117d8c1e4fe57cb135fcefa9d26559dfae26051d27f4632fd2f","family_id":"qf-quadratic-exact-roots","seed":202610034087,"semantic_template":"qf-quadratic-exact-roots/{\"a\":-6,\"b\":-1,\"c\":1,\"domain\":\"reals\",\"operation\":\"exact_roots\"}"},"fields":{"title":"Exact Root Set of a Quadratic Equation","stem":"Over the reals, (-6)x²+(-1)x+(1)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.","answer":{"kind":"expression","value":"{-1/2, 1/3}","explanation":"D=25. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {-1/2, 1/3}. The comparison is exact."},"skills":["Exact Root Set of a Quadratic Equation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0088","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70d1c9d50ec725389d1d891731e64e02f50a4bc00c473ac90fb13ccd58f41a36","family_id":"qf-quadratic-exact-roots","seed":202610034088,"semantic_template":"qf-quadratic-exact-roots/{\"a\":6,\"b\":-4,\"c\":0,\"domain\":\"reals\",\"operation\":\"exact_roots\"}"},"fields":{"title":"Exact Root Set of a Quadratic Equation","stem":"Over the reals, (6)x²+(-4)x+(0)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.","answer":{"kind":"expression","value":"{0/1, 2/3}","explanation":"D=16. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {0/1, 2/3}. The comparison is exact."},"skills":["Exact Root Set of a Quadratic Equation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0089","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"232d865f87f7622a30eb218c7362269131731715af8d328a0f4751e49f60eb7e","family_id":"qf-quadratic-exact-roots","seed":202610034089,"semantic_template":"qf-quadratic-exact-roots/{\"a\":-6,\"b\":9,\"c\":-3,\"domain\":\"reals\",\"operation\":\"exact_roots\"}"},"fields":{"title":"Exact Root Set of a Quadratic Equation","stem":"Over the reals, (-6)x²+(9)x+(-3)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.","answer":{"kind":"expression","value":"{1/2, 1/1}","explanation":"D=9. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {1/2, 1/1}. The comparison is exact."},"skills":["Exact Root Set of a Quadratic Equation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0090","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a28928b02d62588a206f3c8e2c89cbb3a8192d2c9fa94cd4b0f51ca5f776448","family_id":"qf-quadratic-exact-roots","seed":202610034090,"semantic_template":"qf-quadratic-exact-roots/{\"a\":6,\"b\":0,\"c\":-6,\"domain\":\"reals\",\"operation\":\"exact_roots\"}"},"fields":{"title":"Exact Root Set of a Quadratic Equation","stem":"Over the reals, (6)x²+(0)x+(-6)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.","answer":{"kind":"expression","value":"{-1/1, 1/1}","explanation":"D=144. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {-1/1, 1/1}. The comparison is exact."},"skills":["Exact Root Set of a Quadratic Equation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0091","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9d7fdff33d857da90c0913dcb642b7b71edc29b74b13e3981c2414b7fa9dab0","family_id":"qf-quadratic-exact-roots","seed":202610034091,"semantic_template":"qf-quadratic-exact-roots/{\"a\":-6,\"b\":5,\"c\":6,\"domain\":\"reals\",\"operation\":\"exact_roots\"}"},"fields":{"title":"Exact Root Set of a Quadratic Equation","stem":"Over the reals, (-6)x²+(5)x+(6)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.","answer":{"kind":"expression","value":"{-2/3, 3/2}","explanation":"D=169. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {-2/3, 3/2}. The comparison is exact."},"skills":["Exact Root Set of a Quadratic Equation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0092","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6af97392c410ca116445191918e7d47366961e88ab5ab431ab88bfaa80e0c01e","family_id":"qf-quadratic-exact-roots","seed":202610034092,"semantic_template":"qf-quadratic-exact-roots/{\"a\":6,\"b\":-10,\"c\":-4,\"domain\":\"reals\",\"operation\":\"exact_roots\"}"},"fields":{"title":"Exact Root Set of a Quadratic Equation","stem":"Over the reals, (6)x²+(-10)x+(-4)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.","answer":{"kind":"expression","value":"{-1/3, 2/1}","explanation":"D=196. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {-1/3, 2/1}. The comparison is exact."},"skills":["Exact Root Set of a Quadratic Equation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0093","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f044794d269f2775ccb291c2b1ca69ddc087374ea13f5f21a66bf6747aabaadb","family_id":"qf-quadratic-exact-roots","seed":202610034093,"semantic_template":"qf-quadratic-exact-roots/{\"a\":-6,\"b\":15,\"c\":0,\"domain\":\"reals\",\"operation\":\"exact_roots\"}"},"fields":{"title":"Exact Root Set of a Quadratic Equation","stem":"Over the reals, (-6)x²+(15)x+(0)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.","answer":{"kind":"expression","value":"{0/1, 5/2}","explanation":"D=225. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {0/1, 5/2}. The comparison is exact."},"skills":["Exact Root Set of a Quadratic Equation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0094","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7949383c1a1414acaefdcff5cfb8e8c553cef789a4879bfed369e743edca7992","family_id":"qf-quadratic-exact-roots","seed":202610034094,"semantic_template":"qf-quadratic-exact-roots/{\"a\":6,\"b\":-20,\"c\":6,\"domain\":\"reals\",\"operation\":\"exact_roots\"}"},"fields":{"title":"Exact Root Set of a Quadratic Equation","stem":"Over the reals, (6)x²+(-20)x+(6)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.","answer":{"kind":"expression","value":"{1/3, 3/1}","explanation":"D=256. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {1/3, 3/1}. The comparison is exact."},"skills":["Exact Root Set of a Quadratic Equation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0095","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5dec35b4499119f3881b77fc639f975014cf9e825e4df88aaa957def0ac59853","family_id":"qf-quadratic-exact-roots","seed":202610034095,"semantic_template":"qf-quadratic-exact-roots/{\"a\":-6,\"b\":25,\"c\":-14,\"domain\":\"reals\",\"operation\":\"exact_roots\"}"},"fields":{"title":"Exact Root Set of a Quadratic Equation","stem":"Over the reals, (-6)x²+(25)x+(-14)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.","answer":{"kind":"expression","value":"{2/3, 7/2}","explanation":"D=289. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {2/3, 7/2}. The comparison is exact."},"skills":["Exact Root Set of a Quadratic Equation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0096","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1746bc6b9154df787e4ff4b8ba70b2333366adb7c80f209639af00dc9a98311b","family_id":"qf-quadratic-exact-roots","seed":202610034096,"semantic_template":"qf-quadratic-exact-roots/{\"a\":6,\"b\":-30,\"c\":24,\"domain\":\"reals\",\"operation\":\"exact_roots\"}"},"fields":{"title":"Exact Root Set of a Quadratic Equation","stem":"Over the reals, (6)x²+(-30)x+(24)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.","answer":{"kind":"expression","value":"{1/1, 4/1}","explanation":"D=324. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {1/1, 4/1}. The comparison is exact."},"skills":["Exact Root Set of a Quadratic Equation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0097","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3814087bc5c8e3f6411692843738de5b5a36d5423d11a47c5bcb023e8dd086a4","family_id":"qf-quadratic-exact-roots","seed":202610034097,"semantic_template":"qf-quadratic-exact-roots/{\"a\":-6,\"b\":21,\"c\":27,\"domain\":\"reals\",\"operation\":\"exact_roots\"}"},"fields":{"title":"Exact Root Set of a Quadratic Equation","stem":"Over the reals, (-6)x²+(21)x+(27)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.","answer":{"kind":"expression","value":"{-1/1, 9/2}","explanation":"D=1089. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {-1/1, 9/2}. The comparison is exact."},"skills":["Exact Root Set of a Quadratic Equation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0098","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"806983a0e2671b2e543f069b7660e9655ceb66c1bf659be9e01bedd8d4db173c","family_id":"qf-quadratic-exact-roots","seed":202610034098,"semantic_template":"qf-quadratic-exact-roots/{\"a\":6,\"b\":-26,\"c\":-20,\"domain\":\"reals\",\"operation\":\"exact_roots\"}"},"fields":{"title":"Exact Root Set of a Quadratic Equation","stem":"Over the reals, (6)x²+(-26)x+(-20)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.","answer":{"kind":"expression","value":"{-2/3, 5/1}","explanation":"D=1156. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {-2/3, 5/1}. The comparison is exact."},"skills":["Exact Root Set of a Quadratic Equation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0099","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"808cffe974115c229a515063adf009cc6d6b8874c2f30e9552c531a53aa022a9","family_id":"qf-quadratic-exact-roots","seed":202610034099,"semantic_template":"qf-quadratic-exact-roots/{\"a\":-6,\"b\":31,\"c\":11,\"domain\":\"reals\",\"operation\":\"exact_roots\"}"},"fields":{"title":"Exact Root Set of a Quadratic Equation","stem":"Over the reals, (-6)x²+(31)x+(11)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.","answer":{"kind":"expression","value":"{-1/3, 11/2}","explanation":"D=1225. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {-1/3, 11/2}. The comparison is exact."},"skills":["Exact Root Set of a Quadratic Equation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r04-quadratic-equation-0100","batch_id":"batch-20261003-r04-quadratic-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"241e74c967ff7bf18a38bfde29ff8ddfbe638223f56e6f614d0099202dae8f9f","family_id":"qf-quadratic-exact-roots","seed":202610034100,"semantic_template":"qf-quadratic-exact-roots/{\"a\":6,\"b\":-36,\"c\":0,\"domain\":\"reals\",\"operation\":\"exact_roots\"}"},"fields":{"title":"Exact Root Set of a Quadratic Equation","stem":"Over the reals, (6)x²+(-36)x+(0)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.","answer":{"kind":"expression","value":"{0/1, 6/1}","explanation":"D=1296. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {0/1, 6/1}. The comparison is exact."},"skills":["Exact Root Set of a Quadratic Equation"],"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."}}