Algebraic identity application — The sum of two numbers is 100 and their difference is 37. What is the difference of their squares? Use (a^2 - b^2) = (a - b)(a + b).

Introduction / Context:
Many problems become trivial once you recall the right algebraic identity. Here, knowing that the difference of squares factors into product of sum and difference allows instant calculation without finding the individual numbers first.

Given Data / Assumptions:

• a + b = 100.
• a - b = 37.
• We need a^2 - b^2.

Concept / Approach:
Use the identity a^2 - b^2 = (a - b) * (a + b). Substitute the given sum and difference directly to compute the result in one step. This is both faster and less error-prone than solving for a and b first.

Step-by-Step Solution:
Write the identity: a^2 - b^2 = (a - b)(a + b).Substitute: (a - b) = 37 and (a + b) = 100.Compute product: 37 * 100 = 3700.Therefore, a^2 - b^2 = 3700.

Verification / Alternative check:
If desired, solve a = (100 + 37)/2 = 68.5 and b = (100 - 37)/2 = 31.5, then compute 68.5^2 - 31.5^2; it will equal 3700 due to the identity, confirming the result.

Why Other Options Are Wrong:

• 37/100/63/2963: None equals (a - b)(a + b) with the given values. Only 3700 is consistent with the identity.

Common Pitfalls:
Adding or subtracting the numbers first to get individual values unnecessarily; forgetting the identity and attempting long multiplication; mixing up sum and difference positions.

Final Answer:
3700