Solve: x² - 5x + 6 = 0

The quadratic equation x² - 5x + 6 = 0 has solutions x = 2 and x = 3 when the trinomial is factored into (x - 2)(x - 3) = 0 and the zero product property is applied to set each factor equal to zero, yielding x - 2 = 0 → x = 2 and x - 3 = 0 → x = 3, both satisfying the original equation upon verification.

A. x = 1, 6

Substituting x = 1 yields 1 - 5 + 6 = 2 ≠ 0; substituting x = 6 yields 36 - 30 + 6 = 12 ≠ 0 both values fail verification completely. The error likely originates from factoring the constant term 6 as 1 × 6 and incorrectly assuming these are the roots without verifying their sum equals the linear coefficient's opposite (-b = 5). Students might recognize that factors of 6 are needed but fail to apply the critical sum condition: for x² + bx + c = 0, roots r and s must satisfy r + s = -b and rs = c. Here, 1 + 6 = 7 ≠ 5 while 1 × 6 = 6 satisfies the product condition revealing partial application of factoring criteria where learners check only the product condition (rs = c) while neglecting the sum condition (r + s = -b). This "product-only" error represents a pervasive deficiency in quadratic factoring where students identify factor pairs of c but omit verification that their sum equals -b, producing mathematically invalid factorizations that appear plausible due to correct constant term reproduction. The complete failure upon substitution reveals how partial criterion satisfaction produces solutions that satisfy neither equation requirement a dangerous error pattern requiring explicit dual-condition verification protocols (both sum and product) before accepting any factor pair as valid.

B. x = 2, 3

This solution set correctly satisfies the equation through multiple verification pathways demonstrating quadratic mastery. Factoring method: x² - 5x + 6 = (x - 2)(x - 3) = 0; zero product property yields x = 2 or x = 3. Verification substitution: x = 2 → 4 - 10 + 6 = 0 ✓; x = 3 → 9 - 15 + 6 = 0 ✓. Quadratic formula: x = [5 ±√(25 - 24)]/2 = [5 ± 1]/2 → x = 3 or x = 2. Sum-product verification: roots sum to 5 (matches -b) and multiply to 6 (matches c), confirming Vieta's formulas. Graphical interpretation: parabola y = x² - 5x + 6 opens upward with vertex at (2.5, -0.25), crossing x-axis at (2,0) and (3,0). This solution demonstrates comprehensive understanding of quadratic solution methods including factoring with sum-product criteria, zero product property application, quadratic formula execution, Vieta's relationship verification, and graphical interpretation integrated competencies essential for physics (projectile motion roots), economics (break-even analysis), engineering (resonance frequency determination), and optimization where accurate root finding determines critical transition points, equilibrium states, and boundary conditions across quantitative modeling contexts requiring precise solution determination for polynomial equations.

C. x = -2, -3

Substituting x = -2 yields 4 + 10 + 6 = 20 ≠ 0; substituting x = -3 yields 9 + 15 + 6 = 30 ≠ 0 both values produce positive results far from zero. The error likely originates from factoring as (x + 2)(x + 3) = x² + 5x + 6, which solves x² + 5x + 6 = 0 rather than the given equation with -5x. Students correctly identified factors of 6 summing to 5 but failed to recognize the negative linear coefficient requires both factors to have negative signs in the binomial factors: (x - r)(x - s) = x² - (r+s)x + rs. The sign error reveals critical deficiency in connecting coefficient signs to factor signs a foundational gap where learners recognize numerical relationships but lack sign management precision during factor construction. Another plausible pathway involves solving x² + 5x + 6 = 0 correctly for roots -2 and -3, then misapplying to this problem through sign blindness during equation reading. The consistent positive evaluation results (20 and 30) reveal how sign errors in factoring produce solutions that not only fail verification but produce values with opposite sign behavior relative to the actual roots highlighting the necessity of explicit sign tracking protocols during factor construction with visual reinforcement showing how negative linear coefficients require negative root signs in the factored form's binomial expressions.

D. x = -1, -6

Substituting x = -1 yields 1 + 5 + 6 = 12 ≠ 0; substituting x = -6 yields 36 + 30 + 6 = 72 ≠ 0 both values fail verification dramatically. The error likely originates from combining the sign error of option C with the factor pair error of option A: using factors 1 and 6 with negative signs to produce roots -1 and -6. Students might have recognized factors of 6 as 1 and 6, noted the negative linear coefficient requires negative roots, but failed to verify that (-1) + (-6) = -7 ≠ 5 (should equal -b = 5 for the sum of roots). This compound error reveals fragmented understanding where learners apply partial criteria (negative signs for negative coefficient, factor pairs of constant) without integrated verification that both sum and product conditions hold simultaneously with correct signs. The dramatic failure magnitudes (12 and 72) indicate solutions producing values increasingly distant from zero as |x| increases consistent with evaluating the quadratic at points left of both actual roots where the parabola's upward opening produces large positive values. This distractor effectively identifies learners with multiple fragmented misconceptions operating simultaneously without cross-verification a severe deficiency requiring explicit stepwise protocols: (1) identify factor pairs of c, (2) test which pair sums to -b with appropriate signs, (3) construct factors accordingly, (4) verify through substitution before accepting solutions.

Conclusion

The solution set x = 2, 3 emerges through rigorous application of factoring methodology requiring both product condition (factors multiply to constant term 6) and sum condition (factors sum to opposite of linear coefficient 5) with appropriate sign management for the negative linear term, verified through substitution confirming both values satisfy the original equation exactly. This problem reinforces critical algebraic competencies essential across mathematical domains: executing quadratic factoring through systematic factor pair enumeration with dual-condition verification (sum and product), applying the zero product property to extract roots from factored form, recognizing the relationship between coefficient signs and root signs through Vieta's formulas, and verifying solutions through direct substitution to catch sign errors or factor pair miscalculations. Mastery of these integrated skills proves indispensable for calculus (finding critical points), physics (determining time of flight in projectile motion), economics (identifying market equilibrium prices), and engineering (solving characteristic equations for system stability) where accurate root determination enables prediction of system behavior at transition points, equilibrium states, and boundary conditions. The distractors strategically target pervasive error patterns including product-only verification without sum checking, sign management deficiencies in factor construction, and compound errors from multiple fragmented misconceptions highlighting the necessity of explicit dual-condition verification protocols with sign tracking before accepting any factor pair, and mandatory substitution verification for every candidate root to catch errors that compromise solution validity in increasingly complex polynomial equation solving contexts.