# Question 28428: Exact Root Set of a Quadratic Equation [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-0078) 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. R0055-28428 · q-20261003-r04-quadratic-equation-0078 ## Original question and givens Over the reals, (6)x²+(32)x+(10)=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+(32)x+(10)=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](https://ocme.evemisslab.com/en/mko/mko-quadratic-equation/) ## Step-by-step derivation S001 · impliesBefore 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 dependenciesRule: source_answer_roles_or_ordered_fields_without_JS · Uses: G001 S002 · calculationBefore Legacy quadratic ORIGINAL b squared := mul(32, 32) After Legacy quadratic ORIGINAL b squared = 1024 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(6, 10) After Legacy quadratic ORIGINAL a times c = 60 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, 60) After Legacy quadratic ORIGINAL four times ac = 240 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(1024, 240) After Legacy quadratic ORIGINAL b squared minus fourac = 784 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 = [6, 32, 10, 784] 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 S007 · calculationBefore Legacy quadratic DERIVED discriminant exact midpoint square0 := mul(392, 392) After Legacy quadratic DERIVED discriminant exact midpoint square0 = 153664 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, S006 S008 · impliesBefore source roles = [Legacy quadratic DERIVED discriminant, 784, 0, [0, 785], 392] After derived = [153664, No, 0, 392] Conditions: none Squaring is monotone on nonnegative integers;maintain low²<=n