# Question 34625: Exact Substitution of a Rational Root Candidate [Return to catalogue page 174](https://ocme.evemisslab.com/en/questions/page/174/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261003-r04-quadratic-equation-foundations/#q-20261003-r04-quadratic-equation-0075) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0068-34625 · q-20261003-r04-quadratic-equation-0075 ## Original question and givens 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? - G001 — 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? ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-quadratic-equation](https://ocme.evemisslab.com/en/mko/mko-quadratic-equation/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-quadratic-root-verification, [a, b, c, domain, operation, p, q]] After derived = [root_verification, reals] Conditions: none Separate source banks:legacy RAW SAFE with unbounded derived BigInts versus BASE Number.isInteger without invented SAFE/million caps.Operations,domains and permissive/exact keys follow THIS bank.Keep ORIGINAL JSON intact. Step identifiers and dependenciesRule: R68_BANK_specific_keys_operations_and_numeric_guards · Uses: G001 S002 · impliesBefore source roles = [2, -16, 24, 6, 1] After derived = [6/1, Yes] Conditions: none FIRST qualify a≠0 and an ALREADY reduced candidate with POSITIVE denominator.Zero can only be0/1.This task checks THIS candidate;No is legal and does not require a nonnegative or square discriminant. Step identifiers and dependenciesRule: Quadratic_ROOT_candidate_ALREADY_reduced_positive_DEN · Uses: G001, S001 S003 · calculationBefore Root verification SOURCE unused discriminant b squared := mul(-16, -16) After Root verification SOURCE unused discriminant b squared = 256 Conditions: none Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Root verification SOURCE discriminant ORIGINAL a times c := mul(2, 24) After Root verification SOURCE discriminant ORIGINAL a times c = 48 Conditions: none Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Root verification SOURCE unused discriminant four a c := mul(4, 48) After Root verification SOURCE unused discriminant four a c = 192 Conditions: none Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Root verification SOURCE unused signed discriminant := sub(256, 192) After Root verification SOURCE unused signed discriminant = 64 Conditions: none Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore Root verification ORIGINAL candidate numerator squared := mul(6, 6) After Root verification ORIGINAL candidate numerator squared = 36 Conditions: none Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore Root verification ORIGINAL candidate denominator squared := mul(1, 1) After Root verification ORIGINAL candidate denominator squared = 1 Conditions: none Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Root verification ORIGINAL a times numerator squared := mul(2, 36) After Root verification ORIGINAL a times numerator squared = 72 Conditions: none Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore Root verification ORIGINAL b times candidate numerator := mul(-16, 6) After Root verification ORIGINAL b times candidate numerator = -96 Conditions: none Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore Root verification ORIGINAL b numerator denominator := mul(-96, 1) After Root verification ORIGINAL b numerator denominator = -96 Conditions: none Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore Root verification ORIGINAL c times denominator squared := mul(24, 1) After Root verification ORIGINAL c times denominator squared = 24 Conditions: none Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore Root verification ORIGINAL all denominator-cleared terms := sum(72, -96, 24) After Root verification ORIGINAL all denominator-cleared terms = 0 Conditions: none Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · impliesBefore source roles = [2, -16, 24, 6, 1, 64] After derived = [0, Yes] Conditions: none Substitute x=p/q into the ORIGINAL equation and multiply by NONZERO q².This is equivalent to a*p²+b*p*q+c*q²=0.Retain ALL three terms,including zeros.The SOURCE computes D but this family does not gate its sign/squareness or infer the complete root set. Step identifiers and dependenciesRule: Quadratic_ROOT_clear_NONzero_denominator_ALL_three_terms · Uses: G001, S013 ## Reverse checks - S014 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof. - S013 · premise_and_result_audit — Recompute Root verification ORIGINAL all denominator-cleared terms with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute Root verification ORIGINAL c times denominator squared with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute Root verification ORIGINAL b numerator denominator with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Root verification ORIGINAL b times candidate numerator with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Root verification ORIGINAL a times numerator squared with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Root verification ORIGINAL candidate denominator squared with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Root verification ORIGINAL candidate numerator squared with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Root verification SOURCE unused signed discriminant with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Root verification SOURCE unused discriminant four a c with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Root verification SOURCE discriminant ORIGINAL a times c with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Root verification SOURCE unused discriminant b squared with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof. - S001 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof. ## Result Yes The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 9b7356ebbaffbf2fdc0dcf26af4992245b383f8b2454b5a9628d254a39fbcfb4 - candidate record SHA-256: e05e36b9895476613eaa3d9c477a0f538c9eb8ab02fd9e9231514bd862315fbf - semantic review receipt SHA-256: e7a54104fa02c235f7591329bd6db8d626dd74bf0fae29c557b1f269c85066ed - question-first witness SHA-256: 8401a3c2064c1275a6a7ea8f00412fa12286cec7b2db6c305ff9dba3724ae229 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261003-r04-quadratic-equation-0075.json)