$1397 \times 1397 = x$

### Concept & Formula When finding the square of a number close to a large base (like 1400), we utilize the algebraic identity for the square of a binomial difference to drastically reduce calculation time. $$(a - b)^2 = a^2 - 2ab + b^2$$ ### Step-by-Step Solution * Rewrite 1397 as a subtraction from a round base: $$1397 = 1400 - 3$$ * Substitute this into our squaring equation: $$(1400 - 3)^2 = (1400)^2 - 2(1400)(3) + (3)^2$$ * Evaluate each component carefully: $$(1400)^2 = 1960000$$ $$2 \times 1400 \times 3 = 8400$$ $$(3)^2 = 9$$ * Perform the final arithmetic: $$1960000 - 8400 + 9 = 1951609$$ ### Exam Strategy & Shortcut Use the unit digit and estimation to eliminate distractors in seconds. The unit digit of $1397$ is $7$, so the unit digit of the square is the unit digit of $7 \times 7 = 49$, which is $9$. This leaves all options in play. Look at the magnitude: $1400^2 = 1960000$. Since $1397$ is slightly less than $1400$, the answer must be slightly less than $1960000$. Option (a) $1951609$ fits perfectly. Option (c) is vastly too large ($18$ million), and (d) is over $2$ million. Option (b) is too close to $1960000$ to account for subtracting $8400$. ### Common Pitfall A major error is adding the $2ab$ term instead of subtracting it when using the $(a-b)^2$ formula, which would yield a much larger incorrect answer. Always respect the minus sign! ### Final Answer **Therefore, the correct answer is 1951609.**