Evaluate the quadratic identity with decimals: 9.75^2 − 2 × 9.75 × 5.75 + 5.75^2. Recognize and apply the square of a difference.

Introduction / Context:This problem directly uses the identity (a − b)^2 = a^2 − 2ab + b^2. Recognizing the pattern among the decimal numbers avoids heavy computation and leads to a quick, exact result.

Given Data / Assumptions:

• a = 9.75, b = 5.75
• Expression: a^2 − 2ab + b^2

Concept / Approach:Identify the expression as (a − b)^2. Then compute the difference (9.75 − 5.75) and square it. This is faster and less error-prone than squaring decimals separately and combining terms.

Step-by-Step Solution:9.75^2 − 2 × 9.75 × 5.75 + 5.75^2 = (9.75 − 5.75)^2.Compute the difference: 9.75 − 5.75 = 4.00.Square the result: 4.00^2 = 16.

Verification / Alternative check:Explicit expansion also yields 16, but the identity-based approach is cleaner.

Why Other Options Are Wrong:

• 13.25, 12.5, 4, 3.625: These stem from arithmetic slips or misapplying identities (e.g., confusing (a − b)^2 with a^2 − b^2).

Common Pitfalls:Attempting to multiply all decimals out manually, leading to copying or rounding errors.

Final Answer:16