$(55.25)^2 - 637.5625 =$ $x$

### Concept & Formula This requires calculating the square of a decimal ending in $0.25$. We can use the square of a sum identity to break down the calculation into simple integer operations. $$(a + b)^2 = a^2 + 2ab + b^2$$ ### Step-by-Step Solution * **Step 1: Calculate $(55.25)^2$** * Rewrite $55.25$ as $(55 + 0.25)$. * Apply the formula with $a = 55$ and $b = 0.25$: $$ (55 + 0.25)^2 = (55)^2 + 2(55)(0.25) + (0.25)^2 $$ * Find $(55)^2$: A number ending in 5 squared is $n(n+1)$ prepended to $25$. So, $5 \times 6 = 30$, making it $3025$. * Find the middle term: $2 \times 55 \times 0.25 = 110 \times 0.25 = \frac{110}{4} = 27.5$. * Find the last term: $(0.25)^2 = 0.0625$. * Add them together: $3025 + 27.5 + 0.0625 = 3052.5625$. * **Step 2: Execute the final equation** * Subtract the given value from our squared result: $$ 3052.5625 - 637.5625 $$ * The $.5625$ decimals perfectly cancel out, leaving a clean integer subtraction. * $3052 - 637 = 2415$. ### Exam Strategy & Shortcut Use fast approximation combined with decimal logic. You know $(55)^2$ is roughly $3000$. Subtracting $\approx 640$ from $3000$ leaves roughly $2360$. Looking at the choices, $2415$ is the only plausible answer. Furthermore, recognizing that $(0.25)^2$ generates a trailing $.0625$ confirms that subtracting $.5625$ will result in a whole number. ### Common Pitfall Getting bogged down in traditional multiplication for $55.25 \times 55.25$. Whenever a number ends in $.25$ or $.75$, splitting it into an integer plus a fraction (like $1/4$ or $3/4$) makes mental math significantly faster. ### Final Answer Therefore, the correct answer is 2415.