Solving \( n^2 + n - 420 = 0 \) using the quadratic formula:

["# How to Solve ( n^2 + n - 420 = 0 ) Using the Quadratic Formula – A Step-by-Step Guide", "Solve quadratic equations with confidence using the quadratic formula, especially with real-world problems like ( n^2 + n - 420 = 0 ). In this article, we’ll walk through the process of solving this equation step-by-step, helping you understand how the quadratic formula works and applying it effectively.", "---", "## What Is the Quadratic Equation?", "A quadratic equation is any equation of the form:", "[\nax^2 + bx + c = 0\n]", "where ( a ), ( b ), and ( c ) are constants and ( a <br/>\ne 0 ). One powerful method to solve such equations is the quadratic formula:", "[\nn = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "This formula provides the exact solutions (roots) of any quadratic equation.", "---", "## Problem: Solve ( n^2 + n - 420 = 0 )", "We begin with the equation:", "[\nn^2 + n - 420 = 0\n]", "Here, comparing with ( ax^2 + bx + c ), we identify:", "- ( a = 1 )\n- ( b = 1 )\n- ( c = -420 )", "---", "## Step 1: Plug the values into the quadratic formula", "Substitute ( a = 1 ), ( b = 1 ), and ( c = -420 ) into the formula:", "[\nn = \frac{-(1) \pm \sqrt{(1)^2 - 4(1)(-420)}}{2(1)}\n]", "---", "## Step 2: Simplify inside the square root (the discriminant)", "Calculate the discriminant ( D = b^2 - 4ac ):", "[\nD = 1^2 - 4(1)(-420) = 1 + 1680 = 1681\n]", "---", "## Step 3: Take the square root of the discriminant", "[\n\sqrt{1681} = 41\n]", "Since ( 41^2 = 1681 ), this square root simplifies cleanly.", "---", "## Step 4: Complete the calculation", "Now substitute back into the formula:", "[\nn = \frac{-1 \pm 41}{2}\n]", "This gives two possible solutions:", "[\nn = \frac{-1 + 41}{2} = \frac{40}{2} = 20\n]", "[\nn = \frac{-1 - 41}{2} = \frac{-42}{2} = -21\n]", "---", "## Step 5: State the final solutions", "The solutions to the equation ( n^2 + n - 420 = 0 ) are:", "[\nn = 20 \quad \ ext{and} \quad n = -21\n]", "---", "## Why This Matters: Real-World Applications", "Quadratic equations model a wide range of real-life scenarios — from projectile motion and profit calculations to geometry problems. Knowing how to solve them using the quadratic formula ensures you can tackle complex problems with precision.", "---", "## Summary: Key Takeaways", "- Always identify coefficients ( a ), ( b ), and ( c ) carefully.\n- Plug them into the quadratic formula precisely.\n- Simplify the discriminant before taking the square root.\n- Don’t forget both signs in ( \pm ), which yield two solutions.\n- Check solutions by substituting back into the original equation.", "---", "## Final Note", "Mastering the quadratic formula opens the door to solving many algebraic challenges. With practice, solving equations like ( n^2 + n - 420 = 0 ) becomes a straightforward, intuitive process. Keep practicing — and soon, quadratics will be one of your strongest algebra tools!", "---", "Keywords: solve quadratic equations, quadratic formula example, solve ( n^2 + n - 420 = 0 ), step-by-step quadratic solution, quadratic equations made easy, real quadratic problems, algebra practice, math homework help.", "---", "Let this guide empower your confidence in quadratic solving — and remember: every equation tells a story, and mathematics helps you discover it!"]