If α and β are the two roots of the quadratic equation 3x^2 − 13x + 14 = 0, what is the exact value of the expression (α / β) + (β / α)?

Introduction / Context:
This question assesses understanding of relationships between roots and coefficients of a quadratic equation. Instead of solving for α and β directly, you can use Vieta's formulas to compute expressions involving α and β, such as α/β + β/α, in terms of the coefficients of the quadratic polynomial.

Given Data / Assumptions:

• α and β are roots of 3x^2 − 13x + 14 = 0.
• The quadratic is 3x^2 − 13x + 14 with real coefficients.
• We must compute (α / β) + (β / α).

Concept / Approach:
Recall that for a quadratic ax^2 + bx + c = 0 with roots α and β, the sum of roots is α + β = −b/a and the product of roots is αβ = c/a. The expression α/β + β/α can be rewritten as (α^2 + β^2)/(αβ). We can find α^2 + β^2 using the identity α^2 + β^2 = (α + β)^2 − 2αβ, and then divide by αβ using the values given by Vieta's formulas.

Step-by-Step Solution:
For the equation 3x^2 − 13x + 14 = 0, identify a = 3, b = −13, c = 14.The sum of roots is α + β = −b/a = −(−13)/3 = 13/3.The product of roots is αβ = c/a = 14/3.Compute α^2 + β^2 using α^2 + β^2 = (α + β)^2 − 2αβ.(α + β)^2 = (13/3)^2 = 169/9, and 2αβ = 2 * (14/3) = 28/3 = 84/9.So α^2 + β^2 = 169/9 − 84/9 = 85/9.Now compute (α / β) + (β / α) = (α^2 + β^2)/(αβ) = (85/9) / (14/3) = (85/9) * (3/14).Simplify: (85 * 3) / (9 * 14) = 255/126 = 85/42 after dividing numerator and denominator by 3.

Verification / Alternative check:
We could solve the quadratic 3x^2 − 13x + 14 = 0 exactly to find α and β, then compute α/β + β/α numerically. The roots are real and can be found using the quadratic formula. Substituting them into the expression yields a numerical value that simplifies to 85/42, confirming the result obtained using Vieta's formulas and algebraic manipulation.

Why Other Options Are Wrong:

• 9 is an integer and would require (α^2 + β^2) = 9αβ, which contradicts the calculated values.
• 65/28, 53/14, and 15/2 result from miscalculations in squaring the sum or subtracting 2αβ, or from dividing by αβ incorrectly.
• Only 85/42 matches the correct transformed expression.

Common Pitfalls:

• Forgetting to square (α + β) correctly or miscomputing (13/3)^2.
• Errors in converting 2αβ into a fraction with the same denominator before subtraction.
• Misapplying Vieta's formulas, such as using c instead of c/a for αβ.

Final Answer:
85/42