If $(73)^2$ is subtracted from the square of a number, the answer so obtained is 5075. What is the number?

### Concept & Formula This problem translates directly into a straightforward algebraic equation involving squares. We isolate the unknown variable's square to solve. $$ x^2 - 73^2 = 5075 $$ ### Step-by-Step Solution * Let the unknown number be $x$. * According to the problem statement: $$ x^2 - 73^2 = 5075 $$ * Calculate the square of 73: $$ 73 \times 73 = 5329 $$ * Substitute this back into the equation: $$ x^2 - 5329 = 5075 $$ * Isolate $x^2$ by adding 5329 to both sides: $$ x^2 = 5075 + 5329 $$ $$ x^2 = 10404 $$ * Find the square root to get $x$: $$ x = \sqrt{10404} $$ * Knowing that $100^2 = 10000$, the answer must be slightly larger. Checking 102: $102 \times 102 = 10404$. Thus, $x = 102$. ### Exam Strategy & Shortcut Use the **Unit Digit Strategy** to solve this incredibly fast without calculating big squares. * The unit digit of $73^2$ is the unit digit of $3 \times 3$, which is **9**. * The equation is $x^2 - (...9) = (...5)$. * This means the unit digit of $x^2$ must be **4** (because a number ending in 4, like 14, minus 9 equals 5). * Look at the options: (a) $96^2$ ends in 6 ($6 \times 6 = 36$). (b) $98^2$ ends in 4 ($8 \times 8 = 64$). (c) $102^2$ ends in 4 ($2 \times 2 = 4$). (d) $106^2$ ends in 6 ($6 \times 6 = 36$). * It must be 98 or 102. Since $73^2 \approx 5000$, and $5000 + 5075 \approx 10000$, the root must be slightly above 100. Hence, 102 is the logical choice. ### Common Pitfall A common error is making arithmetic mistakes when manually calculating $73 \times 73$. Relying on unit digits and approximations provides a much safer cross-check mechanism. ### Final Answer **Therefore, the correct answer is 102.**