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Question 28429: Exact Root Set of a Quadratic Equation

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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R0055-28429 · q-20261003-r04-quadratic-equation-0079

Original question and givens

Over the reals, (-6)x²+(-27)x+(0)=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, (-6)x²+(-27)x+(0)=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.

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(-27, -27)

After

Legacy quadratic ORIGINAL b squared = 729

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(-6, 0)

After

Legacy quadratic ORIGINAL a times c = 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, S002

S004 · calculation

Before

Legacy quadratic ORIGINAL four times ac := mul(4, 0)

After

Legacy quadratic ORIGINAL four times ac = 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, S003

S005 · calculation

Before

Legacy quadratic ORIGINAL b squared minus fourac := sub(729, 0)

After

Legacy quadratic ORIGINAL b squared minus fourac = 729

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 = [-6, -27, 0, 729]

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 dependencies

Rule: Legacy_quadratic_GENERAL_complete_square_iff_DISTINCT_real_roots · Uses: G001, S005

S007 · calculation

Before

Legacy quadratic DERIVED discriminant exact midpoint square0 := mul(365, 365)

After

Legacy quadratic DERIVED discriminant exact midpoint square0 = 133225

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 · implies

Before

source roles = [Legacy quadratic DERIVED discriminant, 729, 0, [0, 730], 365]

After

derived = [133225, No, 0, 365]

Conditions: none

Squaring is monotone on nonnegative integers;maintain low²<=n<high².The midpoint is BigInt floor((low+high)/2),not floating sqrt,and strictly shrinks the integer bracket.Derived values do not inherit SAFE bounds.

Step identifiers and dependencies

Rule: Legacy_isqrt_EACH_BigInt_integer_midpoint_bracket · Uses: G001, S007

S009 · calculation

Before

Legacy quadratic DERIVED discriminant exact midpoint square1 := mul(182, 182)

After

Legacy quadratic DERIVED discriminant exact midpoint square1 = 33124

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, S008

S010 · implies

Before

source roles = [Legacy quadratic DERIVED discriminant, 729, 1, [0, 365], 182]

After

derived = [33124, No, 0, 182]

Conditions: none

Squaring is monotone on nonnegative integers;maintain low²<=n<high².The midpoint is BigInt floor((low+high)/2),not floating sqrt,and strictly shrinks the integer bracket.Derived values do not inherit SAFE bounds.

Step identifiers and dependencies

Rule: Legacy_isqrt_EACH_BigInt_integer_midpoint_bracket · Uses: G001, S009

S011 · calculation

Before

Legacy quadratic DERIVED discriminant exact midpoint square2 := mul(91, 91)

After

Legacy quadratic DERIVED discriminant exact midpoint square2 = 8281

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 · implies

Before

source roles = [Legacy quadratic DERIVED discriminant, 729, 2, [0, 182], 91]

After

derived = [8281, No, 0, 91]

Conditions: none

Squaring is monotone on nonnegative integers;maintain low²<=n<high².The midpoint is BigInt floor((low+high)/2),not floating sqrt,and strictly shrinks the integer bracket.Derived values do not inherit SAFE bounds.

Step identifiers and dependencies

Rule: Legacy_isqrt_EACH_BigInt_integer_midpoint_bracket · Uses: G001, S011

S013 · calculation

Before

Legacy quadratic DERIVED discriminant exact midpoint square3 := mul(45, 45)

After

Legacy quadratic DERIVED discriminant exact midpoint square3 = 2025

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 · implies

Before

source roles = [Legacy quadratic DERIVED discriminant, 729, 3, [0, 91], 45]

After

derived = [2025, No, 0, 45]

Conditions: none

Squaring is monotone on nonnegative integers;maintain low²<=n<high².The midpoint is BigInt floor((low+high)/2),not floating sqrt,and strictly shrinks the integer bracket.Derived values do not inherit SAFE bounds.

Step identifiers and dependencies

Rule: Legacy_isqrt_EACH_BigInt_integer_midpoint_bracket · Uses: G001, S013

S015 · calculation

Before

Legacy quadratic DERIVED discriminant exact midpoint square4 := mul(22, 22)

After

Legacy quadratic DERIVED discriminant exact midpoint square4 = 484

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, S014

S016 · implies

Before

source roles = [Legacy quadratic DERIVED discriminant, 729, 4, [0, 45], 22]

After

derived = [484, Yes, 22, 45]

Conditions: none

Squaring is monotone on nonnegative integers;maintain low²<=n<high².The midpoint is BigInt floor((low+high)/2),not floating sqrt,and strictly shrinks the integer bracket.Derived values do not inherit SAFE bounds.

Step identifiers and dependencies

Rule: Legacy_isqrt_EACH_BigInt_integer_midpoint_bracket · Uses: G001, S015

S017 · calculation

Before

Legacy quadratic DERIVED discriminant exact midpoint square5 := mul(33, 33)

After

Legacy quadratic DERIVED discriminant exact midpoint square5 = 1089

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, S016

S018 · implies

Before

source roles = [Legacy quadratic DERIVED discriminant, 729, 5, [22, 45], 33]

After

derived = [1089, No, 22, 33]

Conditions: none

Squaring is monotone on nonnegative integers;maintain low²<=n<high².The midpoint is BigInt floor((low+high)/2),not floating sqrt,and strictly shrinks the integer bracket.Derived values do not inherit SAFE bounds.

Step identifiers and dependencies

Rule: Legacy_isqrt_EACH_BigInt_integer_midpoint_bracket · Uses: G001, S017

S019 · calculation

Before

Legacy quadratic DERIVED discriminant exact midpoint square6 := mul(27, 27)

After

Legacy quadratic DERIVED discriminant exact midpoint square6 = 729

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, S018

S020 · implies

Before

source roles = [Legacy quadratic DERIVED discriminant, 729, 6, [22, 33], 27]

After

derived = [729, Yes, 27, 33]

Conditions: none

Squaring is monotone on nonnegative integers;maintain low²<=n<high².The midpoint is BigInt floor((low+high)/2),not floating sqrt,and strictly shrinks the integer bracket.Derived values do not inherit SAFE bounds.

Step identifiers and dependencies

Rule: Legacy_isqrt_EACH_BigInt_integer_midpoint_bracket · Uses: G001, S019

S021 · calculation

Before

Legacy quadratic DERIVED discriminant exact midpoint square7 := mul(30, 30)

After

Legacy quadratic DERIVED discriminant exact midpoint square7 = 900

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, S020

S022 · implies

Before

source roles = [Legacy quadratic DERIVED discriminant, 729, 7, [27, 33], 30]

After

derived = [900, No, 27, 30]

Conditions: none

Squaring is monotone on nonnegative integers;maintain low²<=n<high².The midpoint is BigInt floor((low+high)/2),not floating sqrt,and strictly shrinks the integer bracket.Derived values do not inherit SAFE bounds.

Step identifiers and dependencies

Rule: Legacy_isqrt_EACH_BigInt_integer_midpoint_bracket · Uses: G001, S021

S023 · calculation

Before

Legacy quadratic DERIVED discriminant exact midpoint square8 := mul(28, 28)

After

Legacy quadratic DERIVED discriminant exact midpoint square8 = 784

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, S022

S024 · implies

Before

source roles = [Legacy quadratic DERIVED discriminant, 729, 8, [27, 30], 28]

After

derived = [784, No, 27, 28]

Conditions: none

Squaring is monotone on nonnegative integers;maintain low²<=n<high².The midpoint is BigInt floor((low+high)/2),not floating sqrt,and strictly shrinks the integer bracket.Derived values do not inherit SAFE bounds.

Step identifiers and dependencies

Rule: Legacy_isqrt_EACH_BigInt_integer_midpoint_bracket · Uses: G001, S023

S025 · calculation

Before

Legacy quadratic DERIVED discriminant terminal root square := mul(27, 27)

After

Legacy quadratic DERIVED discriminant terminal root square = 729

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, S024

S026 · calculation

Before

Legacy quadratic DERIVED discriminant terminal successor square := mul(28, 28)

After

Legacy quadratic DERIVED discriminant terminal successor square = 784

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, S025

S027 · implies

Before

source roles = [Legacy quadratic DERIVED discriminant, 729, 27, 28]

After

derived = [729, 784]

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, S026

S028 · calculation

Before

Legacy quadratic ORIGINAL negated b := sub(0, -27)

After

Legacy quadratic ORIGINAL negated b = 27

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, S027

S029 · calculation

Before

Legacy quadratic ORIGINAL minus numerator := sub(27, 27)

After

Legacy quadratic ORIGINAL minus numerator = 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, S028

S030 · calculation

Before

Legacy quadratic ORIGINAL plus numerator := sum(27, 27)

After

Legacy quadratic ORIGINAL plus numerator = 54

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, S029

S031 · calculation

Before

Legacy quadratic ORIGINAL denominator2a := mul(2, -6)

After

Legacy quadratic ORIGINAL denominator2a = -12

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, S030

S032 · calculation

Before

Legacy quadratic ORIGINAL raw rational branch0 := div(0, -12)

After

Legacy quadratic ORIGINAL raw rational branch0 = 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, S031

S033 · implies

Before

source roles = [0, 0, -12]

After

derived = [0, 1, 0/1]

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, S032

S034 · calculation

Before

Legacy quadratic ORIGINAL raw rational branch1 := div(54, -12)

After

Legacy quadratic ORIGINAL raw rational branch1 = -9/2

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, S033

S035 · implies

Before

source roles = [1, 54, -12]

After

derived = [-9, 2, -9/2]

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, S034

S036 · implies

Before

source roles = [-6, 729, [0/1, -9/2]]

After

derived = [[-9/2, 0/1], {-9/2, 0/1}]

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, S035

Reverse checks

  • S036 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
  • S035 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
  • S034 · premise_and_result_audit — Recompute Legacy quadratic ORIGINAL raw rational branch1 with exact rational arithmetic, not rounded decimals.
  • S033 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
  • S032 · premise_and_result_audit — Recompute Legacy quadratic ORIGINAL raw rational branch0 with exact rational arithmetic, not rounded decimals.
  • S031 · premise_and_result_audit — Recompute Legacy quadratic ORIGINAL denominator2a with exact rational arithmetic, not rounded decimals.
  • S030 · premise_and_result_audit — Recompute Legacy quadratic ORIGINAL plus numerator with exact rational arithmetic, not rounded decimals.
  • S029 · premise_and_result_audit — Recompute Legacy quadratic ORIGINAL minus numerator with exact rational arithmetic, not rounded decimals.
  • S028 · premise_and_result_audit — Recompute Legacy quadratic ORIGINAL negated b with exact rational arithmetic, not rounded decimals.
  • S027 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
  • S026 · premise_and_result_audit — Recompute Legacy quadratic DERIVED discriminant terminal successor square with exact rational arithmetic, not rounded decimals.
  • S025 · premise_and_result_audit — Recompute Legacy quadratic DERIVED discriminant terminal root square with exact rational arithmetic, not rounded decimals.
  • S024 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
  • S023 · premise_and_result_audit — Recompute Legacy quadratic DERIVED discriminant exact midpoint square8 with exact rational arithmetic, not rounded decimals.
  • S022 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
  • S021 · premise_and_result_audit — Recompute Legacy quadratic DERIVED discriminant exact midpoint square7 with exact rational arithmetic, not rounded decimals.
  • S020 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
  • S019 · premise_and_result_audit — Recompute Legacy quadratic DERIVED discriminant exact midpoint square6 with exact rational arithmetic, not rounded decimals.
  • 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 — Recompute Legacy quadratic DERIVED discriminant exact midpoint square5 with exact rational arithmetic, not rounded decimals.
  • S016 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
  • S015 · premise_and_result_audit — Recompute Legacy quadratic DERIVED discriminant exact midpoint square4 with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
  • S013 · premise_and_result_audit — Recompute Legacy quadratic DERIVED discriminant exact midpoint square3 with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
  • S011 · premise_and_result_audit — Recompute Legacy quadratic DERIVED discriminant exact midpoint square2 with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
  • S009 · premise_and_result_audit — Recompute Legacy quadratic DERIVED discriminant exact midpoint square1 with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
  • S007 · premise_and_result_audit — Recompute Legacy quadratic DERIVED discriminant exact midpoint square0 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

{-9/2, 0/1}

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: 23408bc48c56211d155e30fe5cbd1de2298e1291cf44e6de1e8feb7f3f458a71
  • candidate record SHA-256: 01345f867fcf271fddad61754d4e47658835f84b1606fcdbe4c1a98c39c1ac64
  • semantic review receipt SHA-256: 6af367dae24a584f3596525c4953bbc2115343efd68cc9083693eb5c1ef2e683
  • question-first witness SHA-256: 611eb69f3716404b968296a8b0d7d8c61b5ca18129d406f1ff9c7c11e10ffb4e
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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