Question 34607: Exact Substitution of a Rational Root Candidate

Original question and givens

Over the reals, (2)x²+(8)x+(6)=0 has a≠0. Is the reduced rational candidate x=-3/1, with positive denominator, a solution?

• G001 — Over the reals, (2)x²+(8)x+(6)=0 has a≠0. Is the reduced rational candidate x=-3/1, with positive denominator, a solution?

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

source roles = [qf-quadratic-root-verification, [a, b, c, domain, operation, p, q]]

derived = [root_verification, reals]

source roles = [2, 8, 6, -3, 1]

derived = [-3/1, Yes]

Root verification SOURCE unused discriminant b squared := mul(8, 8)

Root verification SOURCE unused discriminant b squared = 64

Root verification SOURCE discriminant ORIGINAL a times c := mul(2, 6)

Root verification SOURCE discriminant ORIGINAL a times c = 12

Root verification SOURCE unused discriminant four a c := mul(4, 12)

Root verification SOURCE unused discriminant four a c = 48

Root verification SOURCE unused signed discriminant := sub(64, 48)

Root verification SOURCE unused signed discriminant = 16

Root verification ORIGINAL candidate numerator squared := mul(-3, -3)

Root verification ORIGINAL candidate numerator squared = 9

Root verification ORIGINAL candidate denominator squared := mul(1, 1)

Root verification ORIGINAL candidate denominator squared = 1

Root verification ORIGINAL a times numerator squared := mul(2, 9)

Root verification ORIGINAL a times numerator squared = 18

Root verification ORIGINAL b times candidate numerator := mul(8, -3)

Root verification ORIGINAL b times candidate numerator = -24

Root verification ORIGINAL b numerator denominator := mul(-24, 1)

Root verification ORIGINAL b numerator denominator = -24

Root verification ORIGINAL c times denominator squared := mul(6, 1)

Root verification ORIGINAL c times denominator squared = 6

Root verification ORIGINAL all denominator-cleared terms := sum(18, -24, 6)

Root verification ORIGINAL all denominator-cleared terms = 0

source roles = [2, 8, 6, -3, 1, 16]

derived = [0, Yes]

Reverse checks

• S014 · 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.
• S013 · premise_and_result_audit — Recompute Root verification ORIGINAL all denominator-cleared terms with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Root verification ORIGINAL c times denominator squared with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Root verification ORIGINAL b numerator denominator with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Root verification ORIGINAL b times candidate numerator with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Root verification ORIGINAL a times numerator squared with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Root verification ORIGINAL candidate denominator squared with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Root verification ORIGINAL candidate numerator squared with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Root verification SOURCE unused signed discriminant with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Root verification SOURCE unused discriminant four a c with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Root verification SOURCE discriminant ORIGINAL a times c with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Root verification SOURCE unused discriminant b squared 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.

Result

Yes

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.