Let p + q = 10 and pq = 5 for non-zero p and q. Evaluate the expression p/q + q/p without finding p and q separately.

Introduction / Context:Expressions like p/q + q/p simplify neatly when rewritten in symmetric form. Using the relationship between (p + q)^2 and p^2 + q^2 lets us compute the result directly from the given sum and product.

Given Data / Assumptions:

• p + q = 10
• pq = 5
• We need p/q + q/p.

Concept / Approach:Rewrite p/q + q/p as (p^2 + q^2)/(pq). Then use the identity (p + q)^2 = p^2 + q^2 + 2pq to find p^2 + q^2. Divide by pq to reach the final value without solving a quadratic.

Step-by-Step Solution:p^2 + q^2 = (p + q)^2 − 2pq = 10^2 − 2*5 = 100 − 10 = 90.Therefore p/q + q/p = (p^2 + q^2)/(pq) = 90 / 5 = 18.

Verification / Alternative check:Optionally, solve t^2 − 10t + 5 = 0 to find p and q, then evaluate p/q + q/p numerically to confirm 18. The identity path is far quicker.

Why Other Options Are Wrong:

• 22, 20, 16, 14: These reflect mistakes in forming p^2 + q^2 or dividing by pq.

Common Pitfalls:Forgetting the 2pq term; directly computing p/q and q/p individually, which is unnecessary and error-prone.

Final Answer:18