Question 28427: Exact Root Set of a Quadratic Equation

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0055-28427 · q-20261003-r04-quadratic-equation-0077

Original question and givens

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.

G001 — 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.

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

Step-by-step derivation

S001 · implies

Before

source roles = [qf-quadratic-exact-roots, exact_roots]

After

derived = [[increasing_reduced_positive_denominator_root_SET_retaining1]]

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(2, 2)

After

Legacy quadratic ORIGINAL b squared = 4

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, -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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

Legacy quadratic ORIGINAL b squared minus fourac := sub(4, 4)

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, 2, -1, 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

S007 · calculation

Before

Legacy quadratic DERIVED discriminant terminal root square := mul(0, 0)

After

Legacy quadratic DERIVED discriminant terminal root square = 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, S006

S008 · calculation

Before

Legacy quadratic DERIVED discriminant terminal successor square := mul(1, 1)

After

Legacy quadratic DERIVED discriminant terminal successor square = 1

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, S007

S009 · implies

Before

source roles = [Legacy quadratic DERIVED discriminant, 0, 0, 1]

After

derived = [0, 1]

Conditions: none

Adjacent integer bounds certify floor sqrt=low;an exact root separately needs low²=n,and the principal root is the single nonnegative branch.

Step identifiers and dependencies

Rule: Legacy_isqrt_terminal_floor_and_single_nonnegative_branch · Uses: G001, S008

S010 · calculation

Before

Legacy quadratic ORIGINAL negated b := sub(0, 2)

After

Legacy quadratic ORIGINAL negated b = -2

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, S009

S011 · calculation

Before

Legacy quadratic ORIGINAL minus numerator := sub(-2, 0)

After

Legacy quadratic ORIGINAL minus numerator = -2

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, S010

S012 · calculation

Before

Legacy quadratic ORIGINAL plus numerator := sum(-2, 0)

After

Legacy quadratic ORIGINAL plus numerator = -2

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, S011

S013 · calculation

Before

Legacy quadratic ORIGINAL denominator2a := mul(2, -1)

After

Legacy quadratic ORIGINAL denominator2a = -2

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, S012

S014 · calculation

Before

Legacy quadratic ORIGINAL raw rational branch0 := div(-2, -2)

After

Legacy quadratic ORIGINAL raw rational branch0 = 1

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, S013

S015 · implies

Before

source roles = [0, -2, -2]

After

derived = [1, 1, 1/1]

Conditions: none

Normalize positive denominator and gcd for EACH original branch.Original2a may be negative;derived numerator/denominator do not inherit SAFE.Integer roots retain p/1.

Step identifiers and dependencies

Rule: Legacy_quadratic_EACH_raw_branch_positive_den_gcd_normalization · Uses: G001, S014

S016 · calculation

Before

Legacy quadratic ORIGINAL raw rational branch1 := div(-2, -2)

After

Legacy quadratic ORIGINAL raw rational branch1 = 1

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, S015

S017 · implies

Before

source roles = [1, -2, -2]

After

derived = [1, 1, 1/1]

Conditions: none

Normalize positive denominator and gcd for EACH original branch.Original2a may be negative;derived numerator/denominator do not inherit SAFE.Integer roots retain p/1.

Step identifiers and dependencies

Rule: Legacy_quadratic_EACH_raw_branch_positive_den_gcd_normalization · Uses: G001, S016

S018 · implies

Before

source roles = [-1, 0, [1/1, 1/1]]

After

derived = [[1/1], {1/1}]

Conditions: none

Sort by exact rational value/cross products,not lexically or by assuming the minus branch is smaller(negative a reverses it).At D0 list the repeated value once;the complete-square IFF also proves no roots omitted.

Step identifiers and dependencies

Rule: Legacy_quadratic_EXACT_numeric_sort_and_D0_set_dedup · Uses: G001, S017

Reverse checks

S018 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
S017 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
S016 · premise_and_result_audit — Recompute Legacy quadratic ORIGINAL raw rational branch1 with exact rational arithmetic, not rounded decimals.
S015 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
S014 · premise_and_result_audit — Recompute Legacy quadratic ORIGINAL raw rational branch0 with exact rational arithmetic, not rounded decimals.
S013 · premise_and_result_audit — Recompute Legacy quadratic ORIGINAL denominator2a with exact rational arithmetic, not rounded decimals.
S012 · premise_and_result_audit — Recompute Legacy quadratic ORIGINAL plus numerator with exact rational arithmetic, not rounded decimals.
S011 · premise_and_result_audit — Recompute Legacy quadratic ORIGINAL minus numerator with exact rational arithmetic, not rounded decimals.
S010 · premise_and_result_audit — Recompute Legacy quadratic ORIGINAL negated b with exact rational arithmetic, not rounded decimals.
S009 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
S008 · premise_and_result_audit — Recompute Legacy quadratic DERIVED discriminant terminal successor square with exact rational arithmetic, not rounded decimals.
S007 · premise_and_result_audit — Recompute Legacy quadratic DERIVED discriminant terminal root square with exact rational arithmetic, not rounded decimals.
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

{1/1}

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: 50b07ab0cada2017f3c0b812b9b3f174e199d89f3666903af5ceb77647375087
candidate record SHA-256: 1c7d91f4ae7ed413144ee120f76ef5ebfada44a547971c458c580d1ef32a9601
semantic review receipt SHA-256: 6af367dae24a584f3596525c4953bbc2115343efd68cc9083693eb5c1ef2e683
question-first witness SHA-256: 611eb69f3716404b968296a8b0d7d8c61b5ca18129d406f1ff9c7c11e10ffb4e
Lean: NOT_RUN · human review: NOT_PERFORMED

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