If a and b are the roots of the quadratic equation P x^2 − Q x + R = 0 with P ≠ 0, then what is the value of (1/a^2) + (1/b^2) + (a/b) + (b/a) in terms of P, Q and R?

Introduction / Context:
This is a higher level algebra question involving symmetric expressions of the roots of a quadratic equation. Rather than computing a and b directly, we use the relationships between the coefficients P, Q, R and the roots. The target expression includes a mixture of reciprocals and ratios of a and b, which can be simplified using standard identities.

Given Data / Assumptions:

• a and b are roots of P x^2 − Q x + R = 0, with P ≠ 0.
• Thus a + b = Q / P and ab = R / P.
• We must find E = (1/a^2) + (1/b^2) + (a/b) + (b/a) in terms of P, Q and R.

Concept / Approach:
We treat the expression in two parts. First, we compute 1/a^2 + 1/b^2, and second, we compute a/b + b/a. Both of these parts depend on symmetric combinations of a and b such as a^2 + b^2 and ab. We know that a^2 + b^2 can be written in terms of (a + b)^2 and ab. Then we express everything in terms of Q and P, and finally in terms of P, Q and R. At the end we simplify the algebraic fractions to match one of the options.

Step-by-Step Solution:
1. Use the root coefficient relationships: a + b = Q / P and ab = R / P. 2. Compute a^2 + b^2 using (a + b)^2 = a^2 + b^2 + 2ab. 3. So a^2 + b^2 = (a + b)^2 − 2ab = (Q / P)^2 − 2(R / P). 4. Simplify a^2 + b^2: a^2 + b^2 = (Q^2 / P^2) − 2R / P. 5. Next, compute 1/a^2 + 1/b^2 = (a^2 + b^2) / (a^2 b^2). 6. Note that a^2 b^2 = (ab)^2 = (R / P)^2. 7. Therefore 1/a^2 + 1/b^2 = [(Q^2 / P^2) − 2R / P] / (R^2 / P^2). 8. Simplify this fraction: multiply numerator and denominator by P^2 to remove denominators. 9. This gives 1/a^2 + 1/b^2 = (Q^2 − 2PR) / R^2. 10. Now compute a/b + b/a = (a^2 + b^2) / (ab). 11. Substitute a^2 + b^2 = (Q^2 / P^2) − 2R / P and ab = R / P. 12. So a/b + b/a = [(Q^2 / P^2) − 2R / P] / (R / P). 13. Multiply numerator and denominator by P to simplify: a/b + b/a = [(Q^2 / P) − 2R] / R. 14. Bring to a single fraction: a/b + b/a = (Q^2 − 2PR) / (P R). 15. Now sum the two parts: E = (1/a^2 + 1/b^2) + (a/b + b/a). 16. So E = (Q^2 − 2PR) / R^2 + (Q^2 − 2PR) / (P R). 17. Factor (Q^2 − 2PR) out: E = (Q^2 − 2PR) * [1 / R^2 + 1 / (P R)]. 18. Combine the bracket: 1 / R^2 + 1 / (P R) = (P + R) / (P R^2). 19. Therefore, E = (Q^2 − 2PR) * (P + R) / (P R^2). 20. This matches option A exactly.

Verification / Alternative check:
We can confirm the structure of the result by checking units and symmetry. The expression must be symmetric in a and b, and the final formula depends on P, Q, R only through symmetric combinations. Also, if we choose simple numerical values for P, Q, R that make a and b easy to compute, then evaluate the original expression numerically and compare it against the value from option A, they will match. This consistency supports the correctness of our derivation.

Why Other Options Are Wrong:
Option B changes the sign inside (Q^2 ± 2PR), which would correspond to a different combination of squares and products and does not arise from our derivation.
Option C and option D both omit the factor (P + R) or place denominators differently, so they cannot reproduce the expression we found. None of these can be algebraically rearranged to the final simplified form (P + R)(Q^2 − 2PR) / (P R^2).

Common Pitfalls:
Common mistakes include misusing the formula for a^2 + b^2, forgetting that a^2 b^2 = (ab)^2, or mixing up denominators when combining the two partial fractions. It is also easy to drop a factor of P or R during simplification. Writing each step carefully and factoring the common term (Q^2 − 2PR) explicitly helps keep the algebra organised.

Final Answer:
The required expression equals (P + R)(Q^2 − 2PR) / (P R^2).