# Question 28403: Discriminant and Distinct Real Roots [Return to catalogue page 143](https://ocme.evemisslab.com/en/questions/page/143/) [← 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-0028) 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. R0055-28403 · q-20261003-r04-quadratic-equation-0028 ## Original question and givens Over the reals, (-1)x²+(0)x+(1)=0 has a≠0. Find the discriminant D and the number of distinct real roots. - G001 — Over the reals, (-1)x²+(0)x+(1)=0 has a≠0. Find the discriminant D and the number of distinct real roots. ## 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-discriminant, discriminant] After derived = [[discriminant_and_number_of_DISTINCT_real_roots]] Conditions: none List this scalar/label/set/point answer role or mode-specific ordered fields,not an imposed universal tuple.Retain legacy domain/underscore operation and integer fractions with/1;original bilingual snapshots stay unchanged. Step identifiers and dependenciesRule: source_answer_roles_or_ordered_fields_without_JS · Uses: G001 S002 · calculationBefore Legacy quadratic ORIGINAL b squared := mul(0, 0) After Legacy quadratic ORIGINAL b squared = 0 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Legacy quadratic ORIGINAL a times c := mul(-1, 1) After Legacy quadratic ORIGINAL a times c = -1 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Legacy quadratic ORIGINAL four times ac := mul(4, -1) After Legacy quadratic ORIGINAL four times ac = -4 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Legacy quadratic ORIGINAL b squared minus fourac := sub(0, -4) After Legacy quadratic ORIGINAL b squared minus fourac = 4 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore source roles = [-1, 0, 1, 4] After derived = [4a*(a*x^2+b*x+c)=(2a*x+b)^2-D, 2] Conditions: none With a!=0,multiplication by4a is invertible and the equation is equivalent to(2ax+b)²=D.Negative D gives no real root,zero D one value and positive D two DISTINCT values.A repeated root counts once;D is unbounded derived BigInt. Step identifiers and dependenciesRule: Legacy_quadratic_GENERAL_complete_square_iff_DISTINCT_real_roots · Uses: G001, S005 ## Reverse checks - S006 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct. - S005 · premise_and_result_audit — Recompute Legacy quadratic ORIGINAL b squared minus fourac with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Legacy quadratic ORIGINAL four times ac with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Legacy quadratic ORIGINAL a times c with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Legacy quadratic ORIGINAL b squared with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct. ## Result D=4; distinct real roots=2 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: 318339d8f0e36dd1da465eb1d0bf0ad4f8599806d12995c01c74bb5bd25aa76b - candidate record SHA-256: db3bb976c560cfc4902876cb68be610b883cd6cb5a1a8b65c77dc22bec9e825f - semantic review receipt SHA-256: 6af367dae24a584f3596525c4953bbc2115343efd68cc9083693eb5c1ef2e683 - question-first witness SHA-256: 611eb69f3716404b968296a8b0d7d8c61b5ca18129d406f1ff9c7c11e10ffb4e - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261003-r04-quadratic-equation-0028.json)