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Question 28411: Discriminant and Distinct Real Roots

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0055-28411 · q-20261003-r04-quadratic-equation-0036

Original question and givens

Over the reals, (1)x²+(6)x+(9)=0 has a≠0. Find the discriminant D and the number of distinct real roots.

  • G001 — Over the reals, (1)x²+(6)x+(9)=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.

Step-by-step derivation

S001 · implies

Before

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 dependencies

Rule: source_answer_roles_or_ordered_fields_without_JS · Uses: G001

S002 · calculation

Before

Legacy quadratic ORIGINAL b squared := mul(6, 6)

After

Legacy quadratic ORIGINAL b squared = 36

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

Legacy quadratic ORIGINAL a times c := mul(1, 9)

After

Legacy quadratic ORIGINAL a times c = 9

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

Legacy quadratic ORIGINAL four times ac := mul(4, 9)

After

Legacy quadratic ORIGINAL four times ac = 36

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

Legacy quadratic ORIGINAL b squared minus fourac := sub(36, 36)

After

Legacy quadratic ORIGINAL b squared minus fourac = 0

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

source roles = [1, 6, 9, 0]

After

derived = [4a*(a*x^2+b*x+c)=(2a*x+b)^2-D, 1]

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 dependencies

Rule: 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=0; distinct real roots=1

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: ffdedb3d937fd9339a716d1a49e0e5acf93f95ec01820d8b0de7177e68a951df
  • candidate record SHA-256: e7ea14e20cfdbc10e70c9321e67b1e68565fbe2378896aa5d87857328cb6eb6b
  • semantic review receipt SHA-256: 6af367dae24a584f3596525c4953bbc2115343efd68cc9083693eb5c1ef2e683
  • question-first witness SHA-256: 611eb69f3716404b968296a8b0d7d8c61b5ca18129d406f1ff9c7c11e10ffb4e
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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