Solve the quadratic equation: \(n^2 + n - 420 = 0\).

["# Solve the Quadratic Equation: (n^2 + n - 420 = 0)", "Solving quadratic equations is a fundamental skill in algebra, essential for students, educators, and math enthusiasts alike. One common type of problem is solving equations of the form (n^2 + n - 420 = 0). In this comprehensive guide, we’ll walk through how to solve this quadratic equation step by step, explaining the method using the quadratic formula and factoring when applicable. Whether you're learning algebra or looking to refresh your knowledge, this article provides clear, actionable steps to solve (n^2 + n - 420 = 0).", "---", "## What is a Quadratic Equation?", "A quadratic equation is a second-degree polynomial equation in one variable, generally written as:", "[\nax^2 + bx + c = 0\n]", "where (a), (b), and (c) are constants, and (a <br/>\ne 0). The solutions—called roots—can be real or complex and can be found using:", "- Factoring\n- Completing the square\n- The quadratic formula", "Because (n^2 + n - 420 = 0) has integer coefficients and a straightforward factorization, factoring is a powerful and efficient method here.", "---", "## Step-by-Step Solution of (n^2 + n - 420 = 0)", "### Step 1: Identify the coefficients\nRewrite the equation in standard form:", "[\nn^2 + n - 420 = 0\n]", "Here,\n- (a = 1)\n- (b = 1)\n- (c = -420)", "### Step 2: Factor the quadratic expression\nWe seek two numbers that multiply to (a \cdot c = 1 \cdot (-420) = -420) and add up to (b = 1).", "We search for two integers with a product of (-420) and sum (1).", "After testing factor pairs of 420, we find that:", "[\n21 \ imes (-20) = -420 \quad \ ext{and} \quad 21 + (-20) = 1\n]", "Good! So we can write the factored form as:", "[\n(n + 21)(n - 20) = 0\n]", "### Step 3: Apply the zero-product property\nSet each factor equal to zero:", "[\nn + 21 = 0 \quad \Rightarrow \quad n = -21\n]\n[\nn - 20 = 0 \quad \Rightarrow \quad n = 20\n]", "---", "## Final Answer", "The solutions to the equation (n^2 + n - 420 = 0) are:", "[\nn = -21 \quad \ ext{and} \quad n = 20\n]", "---", "## Understanding the Solutions", "- Interpretation: These values of (n) are the points where the quadratic function (f(n) = n^2 + n - 420) crosses the (n)-axis.\n- Graph insights: The parabola opens upward since (a = 1 > 0), so the parabola crosses the x-axis at (n = -21) and (n = 20).\n- Applications: This type of equation appears in physics, economics, and engineering problems involving area, motion, and optimization.", "---", "## Alternative Method: Using the Quadratic Formula", "For completeness, we can also solve (n^2 + n - 420 = 0) using the quadratic formula:", "[\nn = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Plug in (a = 1), (b = 1), (c = -420):", "[\nn = \frac{-1 \pm \sqrt{1^2 - 4(1)(-420)}}{2(1)} = \frac{-1 \pm \sqrt{1 + 1680}}{2} = \frac{-1 \pm \sqrt{1681}}{2}\n]", "Since (\sqrt{1681} = 41), we get:", "[\nn = \frac{-1 + 41}{2} = 20 \quad \ ext{and} \quad n = \frac{-1 - 41}{2} = -21\n]", "Results match our earlier factoring solution.", "---", "## Tips for Factoring Quadratic Equations Quickly", "- List factor pairs of the constant term (|c|)\n- Find the pair whose difference is (|b|)\n- Choose signs so the middle term matches (b)\n- Replace (+n) with ((+) -) or ((- +)) depending on factor signs", "For (n^2 + n - 420), factoring avoids complex computation, making it ideal for quick solutions.", "---", "## Summary", "Solving (n^2 + n - 420 = 0) reveals two real solutions: (n = -21) and (n = 20). Factoring offers a fast path to the answer when the quadratic expression factors neatly. For advanced learners, using the quadratic formula ensures correctness across all types of quadratics. Understanding both methods strengthens algebraic fluency and problem-solving confidence.", "---", "## Frequently Asked Questions (FAQ)", "Q: Can I always factor a quadratic equation?\nA: Not all quadratics factor easily over the integers. If factoring is difficult, the quadratic formula always works.", "Q: What do the roots mean in real-world contexts?\nA: In applications like projectile motion or revenue modeling, roots often represent critical points like time of impact or break-even levels.", "Q: What if the discriminant ((b^2 - 4ac)) is negative?\nA: No real solutions—roots are complex conjugates.", "---", "Mastering how to solve (n^2 + n - 420 = 0) equips you with core algebraic tools. Practice factoring, apply the quadratic formula, and explore real-world uses to deepen your mathematical insight.", "---", "Keywords: Solve quadratic equation, quadratic formula, factoring (n^2 + n - 420), solutions to (n^2 + n = 420), algebra tutorial, quadratic roots, factor two trinomials, quadratic equations explanation."]