Two numbers have sum 24 and product 143. Compute the sum of their squares.

Introduction / Context:
This is a classic identity application: from sum and product of two numbers, determine the sum of their squares quickly.

Given Data / Assumptions:

• Let the numbers be a and b.
• a + b = 24; ab = 143.

Concept / Approach:
Use the identity a^2 + b^2 = (a + b)^2 − 2ab. This avoids solving for a and b explicitly.

Step-by-Step Solution:

Verification / Alternative check:
Solving: the numbers are 11 and 13 (since t^2 − 24t + 143 = 0 → roots 11, 13). Then 11^2 + 13^2 = 121 + 169 = 290, confirming the identity method.

Why Other Options Are Wrong:
296, 295, and 228 do not match the identity result; they typically come from arithmetic slips (e.g., subtracting ab instead of 2ab).

Common Pitfalls:
Forgetting the factor 2 in 2ab; attempting a longer route by solving for the numbers when the identity is faster and safer.

Final Answer:
290