$106 \times 106 - 94 \times 94 = x$

### Concept & Formula This problem asks for the difference between two perfect squares, represented as $a^2 - b^2$. This is one of the most important factorization identities in mathematics and drastically simplifies the arithmetic. $$a^2 - b^2 = (a + b)(a - b)$$ ### Step-by-Step Solution * Identify $a$ and $b$ from the expression: $$a = 106$$ $$b = 94$$ * Apply the difference of squares identity: $$(106)^2 - (94)^2 = (106 + 94)(106 - 94)$$ * Calculate the sum and the difference inside the brackets: $$106 + 94 = 200$$ $$106 - 94 = 12$$ * Multiply the simplified terms together: $$200 \times 12 = 2400$$ ### Exam Strategy & Shortcut Whenever you see $a^2 - b^2$ on an aptitude test, instantly calculate the sum $(a+b)$. Here, $106 + 94 = 200$. This means the final answer must be a multiple of $200$. Any multiple of $200$ MUST end in two zeroes ($00$). Looking at the options, $2400$ and $2000$ are the only possibilities. Since the difference $(a-b)$ is $12$, the leading digits come from $12 \times 2 = 24$. Option (a) is the instant winner. ### Common Pitfall Not recognizing the pattern and manually calculating $106 \times 106$ and $94 \times 94$ before subtracting. While it works, it turns a 5-second problem into a 60-second chore, which is fatal in competitive exams. ### Final Answer **Therefore, the correct answer is 2400.**