$(46)^2 - (x)^2 = 4398 - 3066$

Concept & Formula This problem involves solving a quadratic equation for an unknown variable $x$. You must evaluate the squares and simple arithmetic on both sides to isolate $x^2$. Step-by-Step Solution * The equation is: $$(46)^2 - x^2 = 4398 - 3066$$ * First, simplify the right-hand side (RHS) of the equation: $$4398 - 3066 = 1332$$ * Next, calculate the square of $46$. You can use the formula $(50 - 4)^2 = 2500 - 400 + 16 = 2116$: $$2116 - x^2 = 1332$$ * Rearrange the equation to isolate $x^2$: $$x^2 = 2116 - 1332$$ $$x^2 = 784$$ * Finally, take the square root of both sides to find $x$: $$x = \sqrt{784}$$ $$x = 28$$ Exam Strategy & Shortcut Use the unit digit trick to save time finding the square root. The RHS subtraction $(4398 - 3066)$ ends in $2$. The square of $46$ ends in $6$ (since $6 \times 6 = 36$). So, the equation in terms of unit digits is: $6 - (\text{unit digit of } x^2) = 2$. This implies the unit digit of $x^2$ must be $4$. A square ending in $4$ means the original number $x$ must end in $2$ or $8$. Looking at the options, we have $28$ and $42$. Since $40^2 = 1600$, which is already greater than $784$, $42$ is too large. Thus, the answer must be $28$. Common Pitfall A common error is manually multiplying $46 \times 46$ and making an addition mistake, then getting stuck on an imperfect square subtraction. Final Answer **Therefore, the correct answer is 28.**