Expand and simplify: \( n^2 + n^2 + 2n + 1 = 85 \), \( 2n^2 + 2n + 1 = 85 \).
["Expand and Simplify: Solving ( n^2 + n^2 + 2n + 1 = 85 ) with Clarity and Precision", "When tackling algebraic equations, one of the most essential steps is expanding and simplifying complex expressions to make them easier to solve. In this article, we’ll walk through the process of expanding and simplifying the equation:", "[\nn^2 + n^2 + 2n + 1 = 85\n]", "This equation may look straightforward, but understanding how to manipulate it correctly is key to finding accurate solutions.", "---", "### Step 1: Expand Like Terms", "The left-hand side contains ( n^2 + n^2 ), which are like terms. To expand:", "[\nn^2 + n^2 = 2n^2\n]", "So the equation becomes:", "[\n2n^2 + 2n + 1 = 85\n]", "This simplification makes the equation cleaner and clearer, allowing for easier rearrangement.", "---", "### Step 2: Move All Terms to One Side", "To solve the equation algebraically, we bring all terms to one side to form a standard quadratic equation:", "[\n2n^2 + 2n + 1 - 85 = 0\n]", "Simplifying the constants:", "[\n2n^2 + 2n - 84 = 0\n]", "This is now a simpler quadratic equation ready for solving using factoring, completing the square, or the quadratic formula.", "---", "### Step 3: Simplify Further (Optional)", "Before applying advanced methods, we can divide the entire equation by the greatest common divisor of the coefficients, which is 2:", "[\n\frac{2n^2 + 2n - 84}{2} = 0 \implies n^2 + n - 42 = 0\n]", "This simplified form:", "[\nn^2 + n - 42 = 0\n]", "is often easier to factor or apply the quadratic formula on.", "---", "### Why Expand and Simplify Matters", "- Clarity: Expanding like terms removes ambiguity and organizes the equation logically.\n- Efficiency: Simplifying reduces complexity before applying solving techniques.\n- Accuracy: Fewer errors occur when working with simpler, standardized forms.", "---", "### Final Solution", "Now that the equation is simplified to:", "[\nn^2 + n - 42 = 0\n]", "We can factor or use the quadratic formula. Factoring gives:", "[\n(n + 7)(n - 6) = 0\n]", "So the solutions are:", "[\nn = -7 \quad \ ext{or} \quad n = 6\n]", "Both values satisfy the original problem, but in context (depending on domain requirements), only positive values may be valid.", "---", "### Conclusion", "Expanding and simplifying algebraic expressions is a powerful technique for solving equations cleanly and efficiently. By mastering this step—combining like terms, organizing terms on one side, and reducing complexity—you lay a strong foundation for solving quadratic and higher-degree equations with confidence.", "Remember:\nSimplify first, solve better.", "---", "Keywords: simplify quadratic equation, expand algebraic expressions, solve ( n^2 + n^2 + 2n + 1 = 85 ), step-by-step algebra, quadratic solution, simplify ( 2n^2 + 2n + 1 = 85 )", "---", "Start expanding and simplifying today to unlock faster, clearer math problem-solving!"]
