Express α^2/β + β^2/α in terms of coefficients: Given ax^2 + bx + c = 0 with roots α and β, find α^2/β + β^2/α in terms of a, b, c.

Introduction / Context:
We are asked to write a symmetric-looking expression in the roots of a quadratic in terms of its coefficients. Vieta’s formulas give α + β = −b/a and αβ = c/a. Use algebraic identities to combine α and β into expressions involving these symmetric sums and products only.

Given Data / Assumptions:

• ax^2 + bx + c = 0 with roots α, β
• S = α + β = −b/a
• P = αβ = c/a
• Target: α^2/β + β^2/α

Concept / Approach:
Notice α^2/β + β^2/α = (α^3 + β^3)/(αβ). Also α^3 + β^3 = (α + β)^3 − 3αβ(α + β) = S^3 − 3PS. Therefore the entire expression equals (S^3 − 3PS)/P = S^3/P − 3S. Substitute S and P in terms of a, b, c and simplify carefully.

Step-by-Step Solution:

Verification / Alternative check:
Test with a simple monic quadratic (a = 1) and numeric roots to confirm that both sides match numerically. The identity holds because it ultimately derives from Vieta’s symmetric relations.

Why Other Options Are Wrong:

• They either have incorrect powers or misplace a and b in denominators, leading to dimensionally inconsistent expressions.

Common Pitfalls:
Dropping a power of a in the denominator or forgetting to divide by P after forming S^3 − 3PS. Keep track of a, b, c consistently.

Final Answer:
(3abc − b^3) / (a^2 c)