$(99.75)^2 - 2250.0625 =$ $x$

### Concept & Formula Rather than performing a complex multiplication for $(99.75)^2$, apply the algebraic identity for the square of a difference to calculate it quickly. $$(a - b)^2 = a^2 - 2ab + b^2$$ ### Step-by-Step Solution * **Step 1: Calculate $(99.75)^2$** * Rewrite $99.75$ as $(100 - 0.25)$ to make the math easier. * Apply the formula where $a = 100$ and $b = 0.25$: $$ (100 - 0.25)^2 = (100)^2 - 2(100)(0.25) + (0.25)^2 $$ * Solve the components: * $(100)^2 = 10000$ * $2 \times 100 \times 0.25 = 50$ * $(0.25)^2 = 0.0625$ * Combine them: $10000 - 50 + 0.0625 = 9950.0625$. * **Step 2: Perform the final subtraction** * Substitute the squared value back into the original equation: $$ 9950.0625 - 2250.0625 $$ * The decimal portions ($0.0625$) perfectly cancel each other out. * $9950 - 2250 = 7700$. ### Exam Strategy & Shortcut Look at the options and the numbers in the question. The number you are subtracting ($2250.0625$) ends in $.0625$. This strongly hints that $(99.75)^2$ will also end in $.0625$ (since $0.75^2$ involves similar decimal trailing). Knowing the decimals will cancel out, focus only on estimating the integers. $100^2$ is $10,000$. $10,000 - 2250$ is $7750$. The only option close to this integer is $7700$. ### Common Pitfall Attempting to multiply $99.75 \times 99.75$ manually using long multiplication. This is incredibly time-consuming and highly prone to alignment and addition errors. ### Final Answer Therefore, the correct answer is 7700.