Solving \( n^2 + n - 110 = 0 \) using the quadratic formula, \( n = \frac{-1 \pm \sqrt{1 + 440}}{2} = \frac{-1 \pm 21}{2} \).

["# Solving ( n^2 + n - 110 = 0 ) Using the Quadratic Formula", "Mathematics offers powerful tools to solve equations that may seem difficult at first glance. One such tool is the quadratic formula, a timeless method for finding the roots of any quadratic equation in the standard form ( ax^2 + bx + c = 0 ). In this article, we’ll walk through solving the equation ( n^2 + n - 110 = 0 ) using this formula and explain each step clearly for better understanding.", "---", "## Understanding the Quadratic Formula", "The general form of a quadratic equation is:", "[\nax^2 + bx + c = 0\n]", "The solutions for ( x ) are given by:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "This formula works for any quadratic equation, regardless of whether the roots are integers, fractions, or irrational numbers.", "---", "## Step 1: Identify coefficients ( a ), ( b ), and ( c )", "Given equation:\n[\nn^2 + n - 110 = 0\n]", "We identify:\n- ( a = 1 ) (coefficient of ( n^2 ))\n- ( b = 1 ) (coefficient of ( n ))\n- ( c = -110 ) (constant term)", "---", "## Step 2: Plug into the Quadratic Formula", "[\nn = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Substituting the values:", "[\nn = \frac{-1 \pm \sqrt{1^2 - 4(1)(-110)}}{2(1)} = \frac{-1 \pm \sqrt{1 + 440}}{2}\n]", "---", "## Step 3: Simplify the Discriminant", "Compute the discriminant ( b^2 - 4ac ):", "[\n1 + 440 = 441\n]", "So the equation becomes:", "[\nn = \frac{-1 \pm \sqrt{441}}{2}\n]", "Since ( \sqrt{441} = 21 ), we have:", "[\nn = \frac{-1 \pm 21}{2}\n]", "---", "## Step 4: Find the Two Solutions", "Using the ( \pm ) to calculate both roots:", "- First root (( + )):\n[\nn = \frac{-1 + 21}{2} = \frac{20}{2} = 10\n]", "- Second root (( - )):\n[\nn = \frac{-1 - 21}{2} = \frac{-22}{2} = -11\n]", "---", "## Step 5: Final Answer", "The solutions to the equation ( n^2 + n - 110 = 0 ) are:", "[\n\boxed{n = 10} \quad \ ext{and} \quad \boxed{n = -11}\n]", "---", "## Why This Matters: Real-World Applications and Learning Insight", "Solving quadratic equations is not just academic—quadratics model many natural and engineered phenomena, such as projectile motion, optimization problems, and circuit analysis. Mastering the quadratic formula equips you to tackle these problems confidently. Recognizing patterns in the discriminant helps determine whether roots are real, equal, or imaginary, enhancing your mathematical intuition.", "By applying the formula step-by-step, even complex equations simplify into manageable expressions—turning confusion into clarity.", "---", "Keywords: solve ( n^2 + n - 110 = 0 ), quadratic formula, quadratic equation solutions, math tutorial, discriminant, algebra, real roots quadratic equation, step-by-step solving, ( n = \frac{-1 \pm \sqrt{441}}{2} )", "---", " meta description: Learn how to solve ( n^2 + n - 110 = 0 ) using the quadratic formula. Step-by-step example with full derivation, discriminant calculation, and final solutions presented clearly for students and math learners."]