$(65)^2 - (55)^2 = x$

### Concept & Formula This is another direct application of the fundamental algebraic identity for evaluating the difference of two perfect squares without manually multiplying. $$a^2 - b^2 = (a - b)(a + b)$$ ### Step-by-Step Solution * Identify the components of the given expression where $a = 65$ and $b = 55$. * Apply the identity substitution directly: $$(65)^2 - (55)^2 = (65 - 55)(65 + 55)$$ * Solve the basic arithmetic expressions inside the parentheses: $$65 - 55 = 10$$ $$65 + 55 = 120$$ * Multiply the isolated results together: $$10 \times 120 = 1200$$ ### Exam Strategy & Shortcut For any $a^2 - b^2$ problem where the difference between $a$ and $b$ is exactly 10, the answer is simply $10 \times (a + b)$. You can mentally add $65 + 55 = 120$ and immediately append a zero to get 1200 instantly without writing anything down. ### Common Pitfall Squaring the numbers ending in 5 manually ($4225 - 3025$) is a valid mathematical trick, but it takes significantly more cognitive load and exam time than simply applying the standardized $a^2 - b^2$ formula. ### Final Answer Therefore, the correct answer is **1200**.