$(7.5 \times 7.5 + 37.5 + 2.5 \times 2.5)$ is equal to

### Concept & Formula This expression fits the classic quadratic expansion pattern for the square of a sum. $$a^2 + 2ab + b^2 = (a + b)^2$$ ### Step-by-Step Solution * Let $a = 7.5$ and $b = 2.5$. * Map out the components of the given expression: * $a^2 = 7.5 \times 7.5$ * $b^2 = 2.5 \times 2.5$ * Test the middle term for $2ab$: $2 \times 7.5 \times 2.5 = 15 \times 2.5 = 37.5$. * Since the middle term matches perfectly, rewrite the entire expression as $(a + b)^2$: $$ (7.5 + 2.5)^2 $$ * Simplify inside the parentheses: $$ 7.5 + 2.5 = 10 $$ * Square the result: $$ 10^2 = 100 $$ ### Exam Strategy & Shortcut Whenever you see $a^2 + \text{middle term} + b^2$, check if the middle term is equal to $2ab$. Here, $7.5 \times 2.5 = 18.75$, and doubling it gives $37.5$. Recognizing this lets you instantly convert the expression to $(7.5 + 2.5)^2 = 10^2 = 100$ within seconds. ### Common Pitfall Manually executing the multiplications ($7.5 \times 7.5 = 56.25$ and $2.5 \times 2.5 = 6.25$) and adding them to $37.5$. While this works, it increases computational overhead and the risk of basic addition mistakes. ### Final Answer Therefore, the correct answer is 100.