Try: suppose \( n(n+2) = 210 \), try \( n=13 \): 195, \( n=14 \): 224. No.

["Solving ( n(n+2) = 210 ): Why Trying ( n = 13 ) and ( n = 14 ) Is Not the Right Approach", "When faced with the equation ( n(n+2) = 210 ), many students jump straight into guess-and-check by testing nearby values such as ( n = 13 ) and ( n = 14 ). While this method may seem intuitive, it’s often inefficient and misleading—especially when a precise algebraic solution prevents repeated trial and error.", "### The Equation and Expected Solutions", "Start by expanding the equation:", "[\nn(n+2) = n^2 + 2n = 210\n]", "This leads to the quadratic equation:", "[\nn^2 + 2n - 210 = 0\n]", "To solve, factor or apply the quadratic formula:", "[\nn = \frac{-2 \pm \sqrt{2^2 + 4 \cdot 210}}{2} = \frac{-2 \pm \sqrt{4 + 840}}{2} = \frac{-2 \pm \sqrt{844}}{2}\n]", "Since ( \sqrt{844} \approx 29.05 ),", "[\nn \approx \frac{-2 + 29.05}{2} \approx 13.5 \quad \ ext{and} \quad n \approx \frac{-2 - 29.05}{2} \approx -15.5\n]", "Only the positive, real solution matters here: approximately ( n \approx 13.5 ), indicating no perfect integer solution. Therefore, no integer ( n ) satisfies ( n(n+2) = 210 ) exactly.", "### Why Trying ( n = 13 ) and ( n = 14 ) Falls Short", "You might try plugging in ( n = 13 ):\n[\n13 \ imes 15 = 195 \quad (\ ext{under by 15})\n]", "And ( n = 14 ):\n[\n14 \ imes 16 = 224 \quad (\ ext{over by 14})\n]", "While this trial is helpful for estimating, it is only approximate and doesn’t guarantee correctness for exact solutions — especially when the problem seeks a precise integer root. Relying solely on trial numbers can lead to confusion, especially when no integer satisfies the equation cleanly.", "### Algebraic Insight: Exact Solution ≠ Guessing", "The real lesson here is that not every equation rewards a brute-force guess. Understanding how to solve quadratics algebraically helps:", "- Determines whether solutions exist and their nature (real, integer, positive, etc.)\n- Avoids wasted time testing values that don’t satisfy the exact equation\n- Supports deeper mathematical reasoning, especially in word problems or functional modeling", "### Conclusion", "Solving ( n(n+2) = 210 ) is best done through algebraic manipulation rather than trial checks. It reveals no integer solution, and the approximation via ( n = 13 ) (195) and ( n = 14 ) (224) serves more as an intuitive guide than a solution. Mastering exact solving techniques equips learners to tackle similar problems with clarity and confidence.", "---", "Keywords: solve ( n(n+2) = 210 ), quadratic equation solution, no integer solution, algebraic method, verify guesses, quadratic formula, math problem solving."]