$$\frac{(963 + 476)^2 + (963 - 476)^2}{(963 \times 963 + 476 \times 476)} = x$$

### Concept & Formula This problem is based on a standard algebraic identity that relates the sum of the squares of a binomial sum and difference directly to the sum of their individual squares. $$(a + b)^2 + (a - b)^2 = 2(a^2 + b^2)$$ ### Step-by-Step Solution * Analyze the given fraction and assign variables to the repeating numbers: * Let $a = 963$ * Let $b = 476$ * Rewrite the entire expression using these variables to reveal the structural pattern: $$\frac{(a + b)^2 + (a - b)^2}{(a \times a + b \times b)}$$ * Simplify the denominator to standard exponential notation: $$\frac{(a + b)^2 + (a - b)^2}{a^2 + b^2}$$ * Substitute the algebraic identity into the numerator: $$\frac{2(a^2 + b^2)}{a^2 + b^2}$$ * Cancel out the common $(a^2 + b^2)$ term from both the numerator and the denominator, leaving only the constant. $$2$$ ### Exam Strategy & Shortcut Whenever you encounter a fraction exactly in the form of $\frac{(a+b)^2 + (a-b)^2}{a^2 + b^2}$, the answer will **always** be 2, regardless of how large or complex the values of $a$ and $b$ are. In a competitive exam, do not even read the specific numbers. Identify the structural pattern and instantly mark 2 to save valuable time. ### Common Pitfall The most common mistake is attempting to add and subtract the massive numbers inside the parentheses, squaring the results, and then attempting long division. This wastes minutes and almost guarantees a fatal calculation error. ### Final Answer Therefore, the correct answer is **2**.