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.
-
A(ab − b^2 c) / (2 b^2 c)
-
B(3bc − a^3) / (b^2 c)
-
C(3ac − b^2) / (a^3 c)
-
D(3abc − b^3) / (a^2 c)
-
E(a^3 + b^3) / (abc)
Answer
Correct Answer: (3abc − b^3) / (a^2 c)
Explanation
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:
S = −b/a, P = c/aS^3/P = (−b/a)^3 ÷ (c/a) = (−b^3/a^3) * (a/c) = −b^3/(a^2 c)−3S = −3(−b/a) = 3b/a = (3ab c)/(a^2 c)Sum: [−b^3 + 3abc]/(a^2 c) = (3abc − b^3)/(a^2 c)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)