Simplify $87 \times 87 + 61 \times 61 - 2 \times 87 \times 61$

### Concept & Formula The expression perfectly matches the algebraic identity for the square of a difference. By identifying this pattern, we reduce the complex arithmetic to a simple subtraction and square: $$ (a - b)^2 = a^2 + b^2 - 2ab $$ ### Step-by-Step Solution * Let $a = 87$ and $b = 61$. * The expression is structured as $(87)^2 + (61)^2 - 2 \times 87 \times 61$, corresponding directly to $a^2 + b^2 - 2ab$. * Substitute this expanded form with the compact identity $(a - b)^2$. * Calculate the difference inside the parenthesis: $$ (87 - 61)^2 = (26)^2 $$ * Expand the square of 26 (using $(20 + 6)^2 = 400 + 36 + 240$ if you do not have it memorized): $$ 676 $$ ### Exam Strategy & Shortcut Spotting the minus sign before the $2ab$ term tells you immediately the formula is $(a - b)^2$. A quick shortcut after finding $(87 - 61) = 26$ is to look at the unit digit. The square of a number ending in 6 must also end in 6 because $6 \times 6 = 36$. You can often eliminate multiple multiple-choice options just by checking this final unit digit. ### Common Pitfall Students sometimes confuse the identity formulas and add the numbers $(87 + 61)$ instead of subtracting them, missing the crucial minus sign in front of the $2 \times 87 \times 61$ term. Always double-check the signs in the formula before doing the arithmetic. ### Final Answer Therefore, the correct answer is **676**.