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

["# Solving the Quadratic Equation (n^2 + n - 420 = 0) Using the Quadratic Formula", "Quadratic equations are fundamental in algebra and appear in various scientific, engineering, and economic applications. One common form is (n^2 + n - 420 = 0), which students and professionals alike can solve using the reliable quadratic formula. In this article, we’ll walk through how to solve this equation step-by-step with detailed explanations, making it easy to master quadratic solutions both by hand and via calculator compatibility.", "## Understanding the Quadratic Equation", "The standard form of a quadratic equation is:\n[ an^2 + bn + c = 0 ]\nFor the equation:\n[\nn^2 + n - 420 = 0\n]\nwe identify the coefficients as:\n- (a = 1)\n- (b = 1)\n- (c = -420)", "Using these values, we apply the quadratic formula:\n[\nn = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]\nThis method applies universally to any quadratic expression, and it guarantees real or complex solutions depending on the discriminant ((b^2 - 4ac)).", "## Step-by-Step Solution", "Let’s substitute (a), (b), and (c) into the quadratic formula:", "[\nn = \frac{-(1) \pm \sqrt{(1)^2 - 4(1)(-420)}}{2(1)}\n]", "Simplify inside the square root:", "[\nn = \frac{-1 \pm \sqrt{1 + 1680}}{2}\n]", "[\nn = \frac{-1 \pm \sqrt{1681}}{2}\n]", "Calculate the square root:", "[\n\sqrt{1681} = 41\n]", "So the equation becomes:", "[\nn = \frac{-1 \pm 41}{2}\n]", "Now compute both solutions using the plus and minus signs:", "### First Solution:\n[\nn = \frac{-1 + 41}{2} = \frac{40}{2} = 20\n]", "### Second Solution:\n[\nn = \frac{-1 - 41}{2} = \frac{-42}{2} = -21\n]", "## Final Answer", "The two real solutions to the equation (n^2 + n - 420 = 0) are:\n[\nn = 20 \quad \ ext{and} \quad n = -21\n]", "---", "## Why This Method Works", "The quadratic formula originates from completing the square, a technique used to transform any quadratic expression into a perfect square trinomial. By substituting coefficients (a), (b), and (c), the formula delivers exact roots—whether real and distinct, real and repeated, or complex—ensuring robustness across all quadratic scenarios.", "## Checking the Solutions", "It’s always a good idea to verify roots by substituting back into the original equation:", "For (n = 20):\n[\n(20)^2 + 20 - 420 = 400 + 20 - 420 = 0\n]", "For (n = -21):\n[\n(-21)^2 + (-21) - 420 = 441 - 21 - 420 = 0\n]", "Both values satisfy the equation, confirming correctness.", "## Using a Calculator", "Modern calculators simplify solving quadratics with built-in quadratic functions. Input (n^2 + n - 420) into the quadratic mode, use (a = 1), (b = 1), (c = -420), and compute the roots. This is especially useful for large coefficients—for example, when (c) is much larger in magnitude, as seen here ((-420)).", "## Applications of Solving Quadratics", "Mastering this method opens doors in multiple domains:", "- Physics: Modeling projectile motion and timing events\n- Engineering: Designing parabolic structures and optimizing areas\n- Economics: Analyzing break-even points and maximizing profit\n- Computer Graphics: Generating smooth curves and motion paths", "Understanding how to solve (n^2 + n - 420 = 0) or similar equations builds a strong foundation for tackling real-world modeling problems.", "## Summary", "Solving the quadratic equation (n^2 + n - 420 = 0) using the quadratic formula yields two clean, rational solutions:\n[\nn = 20 \quad \ ext{and} \quad n = -21\n]\nBy following systematic steps—identifying coefficients, substituting carefully, simplifying, and verifying—you can confidently solve similar quadratics, whether by hand, on paper, or with calculators. This technique remains a cornerstone in mathematical problem-solving and practical application across disciplines.", "Keywords:\nquadratic formula, solving (n^2 + n - 420 = 0), quadratic equations, algebra tutorial, step-by-step solution, formula verification, real roots, quadratic applications"]