What is the sum of two consecutive even numbers, the difference of whose squares is $84$?

### Concept & Formula This problem elegantly unravels if you apply the fundamental algebraic identity for the difference of two squares. It directly links the sum, difference, and the squared difference of two variables. $$a^2 - b^2 = (a - b)(a + b)$$ ### Step-by-Step Solution * **Given:** We have two consecutive even numbers. Let the larger be $a$ and the smaller be $b$. * Because they are consecutive even numbers, their difference is always $2$. $$a - b = 2$$ * **Given:** The difference of their squares is $84$. $$a^2 - b^2 = 84$$ * Substitute these values into the difference of squares identity: $$(a - b)(a + b) = 84$$ $$(2)(a + b) = 84$$ * Divide both sides by $2$ to isolate $(a + b)$: $$a + b = \frac{84}{2}$$ $$a + b = 42$$ * The value $(a + b)$ represents the sum of the two numbers, which is exactly what the question asks for. The sum is $42$. ### Exam Strategy & Shortcut Recognize the pattern instantly: $\text{Sum} = \frac{\text{Difference of Squares}}{\text{Difference of Numbers}}$. Since consecutive even numbers always differ by $2$, you simply divide the given difference of squares by $2$. $84 / 2 = 42$. Mark the answer and move on in $3$ seconds. ### Common Pitfall Many students unnecessarily calculate the individual numbers by creating a system of equations: $a - b = 2$ and $a + b = 42$, finding $2a = 44 \Rightarrow a = 22$, and $b = 20$. Then they add $20 + 22$ to get $42$. This redundant step wastes valuable time since the formula already isolated the sum $(a+b)$. ### Final Answer **Therefore, the correct answer is 42.**