Simplify: \( n^2 + n^2 + 2n + 1 = 85 \) → \( 2n^2 + 2n + 1 = 85 \) → \( 2n^2 + 2n - 84 = 0 \).
["How to Simplify and Solve the Equation: ( n^2 + n^2 + 2n + 1 = 85 )", "Solving quadratic equations efficiently is a key skill in algebra, and simplifying expressions before solving can make the process much easier. One clear example is simplifying the equation:", "[\nn^2 + n^2 + 2n + 1 = 85\n]", "### Step-by-Step Simplification", "Start with the original equation:", "[\nn^2 + n^2 + 2n + 1 = 85\n]", "Combine like terms on the left-hand side:", "[\n2n^2 + 2n + 1 = 85\n]", "Now move all terms to one side to set the equation to zero:", "[\n2n^2 + 2n + 1 - 85 = 0\n]", "[\n2n^2 + 2n - 84 = 0\n]", "This is the simplified quadratic equation ready for solving.", "---", "### Why Simplify First?", "Simplifying before solving reduces clutter and avoids mistakes. Combining the (n^2 + n^2) terms immediately reduces the equation from three terms to two, making it easier to factor or apply the quadratic formula later. It also clarifies the structure for both manual calculation and automated solvers.", "---", "### Solving the Simplified Equation", "You now have:", "[\n2n^2 + 2n - 84 = 0\n]", "Divide every term by 2 to simplify further:", "[\nn^2 + n - 42 = 0\n]", "Now factor the quadratic (if possible):", "Look for two numbers that multiply to (-42) and add to (1). These numbers are (7) and (-6):", "[\n(n + 7)(n - 6) = 0\n]", "Set each factor equal to zero:", "[\nn + 7 = 0 \quad \Rightarrow \quad n = -7\n]\n[\nn - 6 = 0 \quad \Rightarrow \quad n = 6\n]", "---", "### Final Answer", "The solutions are:", "[\n\boxed{n = -7 \quad \ ext{or} \quad n = 6}\n]", "---", "### Practical Takeaway", "When faced with an equation like ( n^2 + n^2 + 2n + 1 = 85 ), recognizing that combining like terms simplifies the solving process. Always:", "1. Combine like terms early.\n2. Move constants to the right side.\n3. Simplify coefficients when possible.\n4. Choose an appropriate method—factoring, completing the square, or quadratic formula—based on equation structure.", "Mastering these steps gives you confidence and speed in solving algebraic problems.", "---", "Keywords for SEO: simplify quadratic equation, solve (2n^2 + 2n - 84 = 0), step-by-step algebra, solving linear and quadratic equations, algebraic simplification, factoring quadratic trinomials."]
