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R0068-34533 · q-20261005-advance-20261006-r02-quadratic-function-0008
On the full real domain, let q:R→R be q(x)=(-8)(x−(-5))²+(-3), with a≠0. Evaluate q(-8) exactly.
G001 — On the full real domain, let q:R→R be q(x)=(-8)(x−(-5))²+(-3), with a≠0. Evaluate q(-8) exactly.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.
Before
source roles = [qf-quadratic-function-evaluation, [a, domain, h, k, operation, x]]
After
derived = [quadratic_evaluation, 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.
Rule: R68_BANK_specific_keys_operations_and_numeric_guards · Uses: G001
Before
source roles = [-8, -5, -3]
After
derived = [q(x)=a*(x-h)^2+k, x in R]
Conditions: none
FIRST qualify nonzero a and the FULL real domain.The ORIGINAL function is vertex form,not a*x²+h*x+k.Keep h/k roles unchanged and do not silently replace the domain by a bounded interval.
Rule: Quadratic_FULL_real_vertex_form_nonzero_leading_coefficient · Uses: G001, S001
Before
Quadratic ORIGINAL x minus h := sub(-8, -5)
After
Quadratic ORIGINAL x minus h = -3
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.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
Quadratic ORIGINAL displacement squared := mul(-3, -3)
After
Quadratic ORIGINAL displacement squared = 9
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.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
Quadratic ORIGINAL a times displacement squared := mul(-8, 9)
After
Quadratic ORIGINAL a times displacement 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.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
Quadratic ORIGINAL weighted square plus k := sum(-72, -3)
After
Quadratic ORIGINAL weighted square plus k = -75
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.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
source roles = [-8, -5, -3, -8]
After
derived = [-3, 9, -72, -75]
Conditions: none
Subtract h,square,multiply by SIGNED a,then add k.Do not lose a negative coefficient;at x=h the square is0.This is THIS input’s value,not a sampled global-extremum certificate.
Rule: Quadratic_EVALUATE_all_substitutions_and_signed_a · Uses: G001, S006
S007 · 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.S006 · premise_and_result_audit — Recompute Quadratic ORIGINAL weighted square plus k with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute Quadratic ORIGINAL a times displacement squared with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute Quadratic ORIGINAL displacement squared with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute Quadratic ORIGINAL x minus h 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.-75
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: fe99029ebdbd65b6c291dca33dcdea2923c60d73842492750f9f4f477d4ceddbcandidate record SHA-256: 48c1631fd362f4b598f9ae57bb0fccff7dfcbc4171f60ad5c4d49adb51ec4864semantic review receipt SHA-256: e7a54104fa02c235f7591329bd6db8d626dd74bf0fae29c557b1f269c85066edquestion-first witness SHA-256: 8401a3c2064c1275a6a7ea8f00412fa12286cec7b2db6c305ff9dba3724ae229Lean: NOT_RUN · human review: NOT_PERFORMED