Set \( n(n+2) = 210 \), \( n^2 + 2n - 210 = 0 \), \( n = \frac{-2 \pm \sqrt{4 + 840}}{2} = \frac{-2 \pm \sqrt{844}}{2} \).

["Solving the Equation ( n(n+2) = 210 ): A Step-by-Step Guide", "When faced with a quadratic equation like ( n(n+2) = 210 ), it’s helpful to rewrite it in standard form and solve using algebraic methods. This article explores how to solve ( n(n+2) = 210 ), derives the quadratic equation, and computes its solutions step by step.", "---", "### 1. Rewriting the Equation", "Start with the original equation:", "[\nn(n + 2) = 210\n]", "Expanding the left side:", "[\nn^2 + 2n = 210\n]", "Move all terms to one side to form a standard quadratic equation:", "[\nn^2 + 2n - 210 = 0\n]", "---", "### 2. Applying the Quadratic Formula", "The quadratic equation ( n^2 + 2n - 210 = 0 ) has the form ( ax^2 + bx + c = 0 ), where:", "- ( a = 1 )\n- ( b = 2 )\n- ( c = -210 )", "Use the quadratic formula:", "[\nn = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Substitute the values:", "[\nn = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-210)}}{2(1)} = \frac{-2 \pm \sqrt{4 + 840}}{2} = \frac{-2 \pm \sqrt{844}}{2}\n]", "---", "### 3. Simplifying the Square Root", "Check if ( \sqrt{844} ) can be simplified:", "[\n844 = 4 \ imes 211\n]\n[\n\sqrt{844} = \sqrt{4 \ imes 211} = 2\sqrt{211}\n]", "So the solutions become:", "[\nn = \frac{-2 \pm 2\sqrt{211}}{2}\n]", "Cancel the 2 in numerator and denominator:", "[\nn = -1 \pm \sqrt{211}\n]", "These are the exact solutions to the equation.", "---", "### 4. Approximate Numerical Value", "Since ( \sqrt{211} \approx 14.525 ), the approximate positive solution is:", "[\nn \approx -1 + 14.525 = 13.525\n]", "Note that ( n = -1 - \sqrt{211} \approx -15.525 ) is negative and typically not meaningful in contexts requiring positive ( n ) (e.g., real-world applications).", "---", "### 5. Verifying the Solution", "To confirm:", "Calculate ( n(n+2) ) with ( n = -1 + \sqrt{211} ):", "[\nn + 2 = 1 + \sqrt{211}\n]\n[\nn(n+2) = (-1 + \sqrt{211})(1 + \sqrt{211}) = (-1)(1) + (-1)(\sqrt{211}) + \sqrt{211}(1) + \sqrt{211} \cdot \sqrt{211}\n]\n[\n= -1 - \sqrt{211} + \sqrt{211} + 211 = 210\n]", "This checks out.", "---", "### Conclusion", "Solving ( n(n+2) = 210 ) leads to the quadratic ( n^2 + 2n - 210 = 0 ), whose positive solution is:", "[\nn = -1 + \sqrt{211}\n]", "Understanding this process helps simplify nested expressions and solve similar algebraic problems efficiently. Whether studying algebra or applying equations in real-world modeling, mastering quadratic solutions is essential.", "---", "Keywords:\nSet ( n(n+2) = 210 ), solve quadratic equation, ( n^2 + 2n - 210 = 0 ), quadratic formula, exact solution, approximate value, algebra, math tips, solving quadratics.\nMeta description:\nSolve ( n(n+2) = 210 ) using algebra. Derive the quadratic form ( n^2 + 2n - 210 = 0 ), apply the quadratic formula, and verify the positive solution with exact and approximate values. Ideal for students learning algebra."]