If $a$ and $b$ are positive integers, $a > b$ and $(a + b)^2 - (a - b)^2 > 29$, then the smallest value of $a$ is

### Concept & Formula The difference between the square of a sum and the square of a difference of two variables simplifies directly to four times their product. $$(a + b)^2 - (a - b)^2 = 4ab$$ ### Step-by-Step Solution * **Given:** $(a + b)^2 - (a - b)^2 > 29$, where $a > b$, and both are positive integers. * Substitute the simplified algebraic identity into the given inequality: $$4ab > 29$$ * Divide both sides by 4 to isolate the product $ab$: $$ab > \frac{29}{4}$$ $$ab > 7.25$$ * Since $a$ and $b$ are positive integers, their product $ab$ must be an integer of at least 8. * We need the smallest possible value for $a$, keeping in mind the strict condition that $a > b$. * Let's test the options starting from the smallest integer: * If $a = 3$, the maximum integer value for $b$ (since $a > b$) is 2. The product $ab = 6$. This is not greater than 7.25. * If $a = 4$, the possible integer values for $b$ are 1, 2, or 3. If $b = 2$, $ab = 8$. If $b = 3$, $ab = 12$. Both results are greater than 7.25. * Thus, the smallest valid integer for $a$ that satisfies all conditions is 4. ### Exam Strategy & Shortcut Recognize the $4ab$ expansion immediately. Knowing $4ab > 29$ means $ab \ge 8$. Since $a > b$, $a$ must be strictly greater than $\sqrt{8}$. The square root of 8 is approximately 2.8, so $a$ must be at least 3. Test $a=3 \rightarrow \text{max } b=2 \rightarrow ab=6$ (Fail). Move to $a=4 \rightarrow \text{max } b=3 \rightarrow ab=12$ (Pass). The answer is 4. ### Common Pitfall A frequent mistake is ignoring the condition $a > b$ and assuming $b$ can be any large integer, which might lead one to believe $a$ could be 1 or 2. Always strictly enforce all inequality constraints while testing options. ### Final Answer Therefore, the correct answer is **4**.