Use identities to evaluate a ratio: Given a^2 + b^2 = 234 and ab = 108, compute the value of (a + b)/(a − b). Assume real numbers where the expression is defined.

Introduction / Context:
This item examines the use of algebraic identities to transform sums and differences into expressions of a^2 + b^2 and ab. It is a staple technique for speed-solving symmetric expressions without solving for a and b individually.

Given Data / Assumptions:

• a^2 + b^2 = 234.
• ab = 108.
• We need (a + b)/(a − b), assuming a ≠ b so the denominator is nonzero.

Concept / Approach:
Use (a + b)^2 = a^2 + b^2 + 2ab and (a − b)^2 = a^2 + b^2 − 2ab. Then the square of the desired ratio equals (a + b)^2 / (a − b)^2, letting us avoid computing a and b explicitly. Take the positive root for the standard magnitude unless a sign is specified.

Step-by-Step Solution:
Compute (a + b)^2 = 234 + 2*108 = 234 + 216 = 450.Compute (a − b)^2 = 234 − 2*108 = 234 − 216 = 18.Thus ((a + b)/(a − b))^2 = 450/18 = 25.Therefore (a + b)/(a − b) = 5 (taking the principal value).

Verification / Alternative check:
If needed, one can construct values of a and b satisfying the conditions (e.g., solving a quadratic with sum s and product p) and confirm the ratio equals 5; however, the identity-based route is faster and exact.

Why Other Options Are Wrong:
10 and 8 do not respect the derived ratio; 4 gives a squared ratio of 16, not 25; 6 gives 36, inconsistent with 450/18.

Common Pitfalls:
Forgetting the 2ab term in the identities; attempting to solve for a and b directly; sign confusion about taking the square root.

Final Answer:
5