Suppose p/q = (x + 3)/(x − 3), where x is a real number with x ≠ 3, x ≠ −3, and x ≠ 0. Find the simplified value of: (p^2 + q^2) / (p^2 − q^2) expressed purely in terms of x.

Introduction / Context: This question tests algebraic manipulation using a ratio. When p/q is known, expressions involving p^2 and q^2 can often be rewritten in terms of r = p/q, and then simplified into a clean formula in x.

Given Data / Assumptions:

• p/q = (x + 3)/(x − 3) • x ≠ 3, x ≠ −3, x ≠ 0 (to avoid invalid denominators and division by zero) • Required: (p^2 + q^2)/(p^2 − q^2)

Concept / Approach: Let r = p/q. Then: (p^2 + q^2)/(p^2 − q^2) = (r^2 + 1)/(r^2 − 1), because dividing numerator and denominator by q^2 gives: (p^2/q^2 + 1)/(p^2/q^2 − 1) = (r^2 + 1)/(r^2 − 1). Then substitute r = (x + 3)/(x − 3) and simplify using common denominators and the identities for sum and difference of squares.

Step-by-Step Solution: 1) Let r = p/q = (x + 3)/(x − 3) 2) Convert expression: (p^2 + q^2)/(p^2 − q^2) = (r^2 + 1)/(r^2 − 1) 3) Compute r^2 = (x + 3)^2/(x − 3)^2 4) Evaluate (r^2 + 1)/(r^2 − 1) using common denominator: = [ (x + 3)^2 + (x − 3)^2 ] / [ (x + 3)^2 − (x − 3)^2 ] 5) Expand: (x + 3)^2 = x^2 + 6x + 9, (x − 3)^2 = x^2 − 6x + 9 6) Sum: 2x^2 + 18 = 2(x^2 + 9) 7) Difference: (x^2 + 6x + 9) − (x^2 − 6x + 9) = 12x 8) Final: 2(x^2 + 9)/(12x) = (x^2 + 9)/(6x)

Verification / Alternative check: Test x = 6: r = 9/3 = 3. Then (r^2 + 1)/(r^2 − 1) = (9 + 1)/(9 − 1) = 10/8 = 5/4. Formula gives (36 + 9)/(36) = 45/36 = 5/4. Matches.

Why Other Options Are Wrong: • Using (x^2 − 9) or swapping numerator/denominator comes from incorrect expansion or sign errors in the difference. • 0 cannot be correct because the expression is not identically zero for valid x.

Common Pitfalls: • Forgetting to square r before adding/subtracting 1. • Miscomputing (x + 3)^2 − (x − 3)^2 (it equals 12x, not x^2 − 9).

Final Answer: (x^2 + 9) / (6x)