This expands to \( n^2 - 2n + 1 + n^2 + n^2 + 2n + 1 = 365 \).
["Title: Solve the Algebraic Equation: ( n^2 - 2n + 1 + n^2 + n^2 + 2n + 1 = 365 )", "---", "Introduction\nEver met a math problem that looks complex at first glance but hides a simple solution? This equation demonstrates how combining algebraic expressions step-by-step can unlock a clean, elegant answer. In this article, we’ll explore how to solve the equation:", "[\nn^2 - 2n + 1 + n^2 + n^2 + 2n + 1 = 365\n]", "By breaking it down, simplifying it, and solving for ( n ), we’ll uncover the value that makes this identity true — and why understanding such problems is valuable beyond homework.", "---", "### Step 1: Combine Like Terms\nStart with the left-hand side of the equation:", "[\nn^2 - 2n + 1 + n^2 + n^2 + 2n + 1\n]", "Group the like terms:", "- Quadratic terms: ( n^2 + n^2 + n^2 = 3n^2 )\n- Linear terms: ( -2n + 2n = 0 )\n- Constant terms: ( 1 + 1 = 2 )", "So the equation simplifies to:", "[\n3n^2 + 2 = 365\n]", "---", "### Step 2: Isolate the Quadratic Term\nSubtract 2 from both sides:", "[\n3n^2 = 363\n]", "Divide both sides by 3:", "[\nn^2 = 121\n]", "---", "### Step 3: Solve for ( n )\nTake the square root of both sides:", "[\nn = \pm 11\n]", "Since ( n ) typically represents a positive quantity in algebra problems, the valid solution is:", "[\nn = 11\n]", "---", "### Why This Matters\nThis problem showcases how quadratic expressions — even when starting with deceptively complex terms — reduce to simple, solvable forms. It exemplifies foundational algebra skills used in physics, engineering, and computer science. Understanding such equations helps build logical thinking and problem-solving expertise applicable to higher math disciplines.", "---", "### Final Answer\n[\n\boxed{11}\n]", "---", "Boost Your Math Skills:\nWant to master algebra? Try solving equations like this regularly and explore patterns in quadratic identities. Share this guide with classmates and dive deeper into transforming expressions — clarity always starts from the basics!"]
