Consecutive positive odd integers (sum of squares = 290): Find two consecutive positive odd integers whose squares add up to 290.

Introduction / Context:
Let the odd integers be n and n + 2. Translating the statement “their squares add to 290” yields a quadratic equation in n. Solving it produces the valid positive odd integers. This is a standard modeling step from words to algebra.

Given Data / Assumptions:

• Integers are consecutive and odd: n and n + 2.
• n > 0.
• n^2 + (n + 2)^2 = 290.

Concept / Approach:
Expand, simplify to a quadratic in n, and solve. Discard negative solutions since the integers must be positive.

Step-by-Step Solution:

Verification / Alternative check:
11^2 + 13^2 = 121 + 169 = 290 as required.

Why Other Options Are Wrong:

• 13, 15: 169 + 225 = 394, not 290.
• 9, 11: 81 + 121 = 202, not 290.
• None of these: Incorrect since 11 and 13 satisfy the condition.

Common Pitfalls:
Forgetting that consecutive odds differ by 2 (not 1), or arithmetic mistakes while expanding squares.

Final Answer: