Multiply by 2: \(n(n+1) = 420\).
["# Multiply by 2: Solve (n(n+1) = 420)—A Step-by-Step Guide", "Solving quadratic equations can feel intimidating, but equations like (n(n+1) = 420) are designed to be approachable with the right strategy. In this article, we’ll explore how to solve (n(n+1) = 420), step-by-step, and uncover the value of (n) using simple algebra—perfect for students, teachers, or anyone looking to master foundational math skills.", "## What Is (n(n+1) = 420)?", "The equation (n(n+1) = 420) is a classic example of a quadratic expression in factored form. Expanding it gives:", "[\nn^2 + n = 420\n]", "Rearranging into standard quadratic form:", "[\nn^2 + n - 420 = 0\n]", "Now we have a solvable quadratic equation ready to be tackled.", "---", "## Why This Equation Matters", "Understanding how to solve equations like (n(n+1) = 420) builds confidence in algebraic reasoning. These patterns appear in number theory, project planning, and algorithm design, making them valuable tools beyond the classroom.", "---", "## Step-by-Step Solution: How to Solve (n(n+1) = 420)", "### Step 1: Expand and Rearrange\nStart with:", "[\nn(n+1) = 420\n]", "Expand the left side:", "[\nn^2 + n = 420\n]", "Bring all terms to one side:", "[\nn^2 + n - 420 = 0\n]", "### Step 2: Factor the Quadratic", "We’re seeking two numbers that multiply to (-420) and add to (1). After testing factor pairs (e.g., 21 × 20 = 420), we find:", "[\n(n + 21)(n - 20) = 0\n]", "### Step 3: Apply the Zero Product Property", "Set each factor equal to zero:", "[\nn + 21 = 0 \quad \Rightarrow \quad n = -21\n]\n[\nn - 20 = 0 \quad \Rightarrow \quad n = 20\n]", "---", "## Interpreting the Solutions", "Since (n) represents a real-world quantity (like time, inventory, or count), (n = -21) is not meaningful. The valid solution is:", "[\nn = 20\n]", "Check by substituting back:", "[\n20(20+1) = 20 \ imes 21 = 420\n]", "✅ The equation holds true.", "---", "## Tips for Solving Similar Equations", "- Rewrite in Standard Form: Always move all terms to one side to form (ax^2 + bx + c = 0).\n- Look for Patterns: Recognize simple factor pairs (e.g., 20 × 21 = 420).\n- Verify Solutions: Plug answers back into the original equation.\n- Consider Context: Keep in mind whether your variable represents a physical quantity (non-negative) or a purely mathematical one.", "---", "## Final Notes", "Solving (n(n+1) = 420) teaches foundational algebraic techniques that break down complex problems into manageable steps. Whether you’re curious about quadratic identities, practicing problem-solving, or preparing for math exams, mastering equations like this strengthens your mathematical toolkit.", "Memorize the key takeaway:\nTo solve (n(n+1) = 420), rewrite it as (n^2 + n - 420 = 0), then factor into ((n + 21)(n - 20) = 0), yielding (n = 20).", "---", "## Want to Practice More?", "Try similar problems like:\n- (n(n-1) = 306)\n- (x(x+2) = 305)\nThese reinforce your ability to manipulate and solve quadratic expressions efficiently.", "Unlock the power of algebra—your journey starts with a single equation.", "---", "Keywords for SEO:\n- Solve (n(n+1) = 420)\n- Quadratic equation solution\n- Algebraic step-by-step guide\n- Factor quadratic (n^2 + n - 420 = 0)\n- Math tips for solving quadratics\n- Learn how to solve (n(n+1) = 420)\n- Step-by-step math problems", "Meta Description:\nMaster the equation (n(n+1) = 420) with our step-by-step guide. Learn to factor, solve, and verify quadratic expressions using simple algebra—perfect for students and math enthusiasts."]
