Solve the quadratic equation: n² + n - 420 = 0.

["# Solve the Quadratic Equation: n² + n - 420 = 0", "Whether you're a student, a math enthusiast, or tackling real-world problems, knowing how to solve quadratic equations is essential. One common challenge is solving equations of the form n² + n - 420 = 0. In this article, we’ll walk you through the step-by-step process of solving this quadratic equation using both traditional algebra methods and the quadratic formula. Plus, we’ll explore practical applications where this type of equation appears.", "---", "## What Is a Quadratic Equation?", "A quadratic equation has the standard form:", "[\nan^2 + bn + c = 0\n]", "where ( a ), ( b ), and ( c ) are constants, and ( a <br/>\neq 0 ). In our equation:", "[\nn^2 + n - 420 = 0\n]", "we identify:\n- ( a = 1 )\n- ( b = 1 )\n- ( c = -420 )", "Quadratic equations can be solved using:", "- Factoring (when possible)\n- The quadratic formula\n- Completing the square", "Let’s focus on the most accessible and widely used method—factoring and the quadratic formula—to solve ( n^2 + n - 420 = 0 ).", "---", "## Step 1: Factoring the Quadratic Expression", "Factoring involves expressing the quadratic as a product of two binomials. Our goal is to rewrite:", "[\nn^2 + n - 420 = (n + m)(n - p) = 0\n]", "Expanding this gives:", "[\nn^2 + (m - p)n - mp\n]", "By comparing coefficients:\n- ( m - p = 1 ) (coefficient of ( n ))\n- ( -mp = -420 ) → ( mp = 420 ) (constant term)", "We need two numbers that multiply to 420 and differ by 1.", "After testing factor pairs of 420, we find:", "- ( 21 \ imes 20 = 420 )\n- ( 21 - 20 = 1 )", "So, we write:", "[\nn^2 + n - 420 = (n + 21)(n - 20) = 0\n]", "Therefore, the equation becomes:", "[\n(n + 21)(n - 20) = 0\n]", "---", "## Step 2: Applying the Zero Product Property", "If a product equals zero, then one of the factors must be zero:", "[\nn + 21 = 0 \quad \ ext{or} \quad n - 20 = 0\n]", "Solving these gives:", "[\nn = -21 \quad \ ext{or} \quad n = 20\n]", "---", "## Step 3: Verifying the Solutions", "Let’s plug the values back into the original equation to confirm:", "- For ( n = 20 ):\n ( 20^2 + 20 - 420 = 400 + 20 - 420 = 0 ) ✅\n- For ( n = -21 ):\n ( (-21)^2 + (-21) - 420 = 441 - 21 - 420 = 0 ) ✅", "Both solutions satisfy the equation.", "---", "## Alternative Method: Using the Quadratic Formula", "The quadratic formula offers a universal solution:", "[\nn = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "For ( n^2 + n - 420 = 0 ), recall:\n( a = 1 ), ( b = 1 ), ( c = -420 )", "Calculate the discriminant:", "[\n\Delta = b^2 - 4ac = 1^2 - 4(1)(-420) = 1 + 1680 = 1681\n]", "Take the square root:", "[\n\sqrt{1681} = 41\n]", "Apply the formula:", "[\nn = \frac{-1 \pm 41}{2}\n]", "So:", "[\nn = \frac{-1 + 41}{2} = \frac{40}{2} = 20\n\quad \ ext{and} \quad\nn = \frac{-1 - 41}{2} = \frac{-42}{2} = -21\n]", "Same solutions confirmed: ( n = 20 ) and ( n = -21 ).", "---", "## Real-World Applications of Quadratic Equations", "Quadratic equations like ( n^2 + n - 420 = 0 ) appear in several practical scenarios:", "- Projectile motion: Calculating the height or time of flight when modeled with quadratic relationships.\n- Area problems: Finding unknown dimensions when area is given in terms of one side.\n- Financial math: Modeling profit, cost, or revenue curves where quadratic behavior emerges.\n- Engineering and architecture: Designing parabolic structures or optimizing load-bearing curves.", "---", "## Key Takeaways", "- The equation ( n^2 + n - 420 = 0 ) factors neatly into ( (n + 21)(n - 20) = 0 )\n- Solutions are ( n = -21 ) and ( n = 20 ), verified via factoring and the quadratic formula\n- Understanding quadratics enhances problem-solving across sciences, engineering, and economics\n- Practice with factoring, estimation, and the quadratic formula builds confidence in handling similar equations", "---", "## Summary", "Solving a quadratic equation step-by-step not only reveals the mathematical solution but also deepens conceptual understanding. For ( n^2 + n - 420 = 0 ), both factoring and the quadratic formula confirm the solutions:", "[\n\boxed{n = -21 \quad} \ ext{and} \quad \boxed{n = 20}\n]", "Whether solving problems alone or collaboratively, mastering these techniques opens doors to expert-level math fluency.", "---", "Keywords: solve quadratic equation, solve n² + n - 420 = 0, quadratic formula, factoring, algebra, quadratic solutions, math tutorial, equation problems.\nAlso search for: how to solve quadratic equations, step-by-step quadratic equations, quadratic formula examples, factor quadratic equation, real-world quadratic applications."]