Equation: \( n^2 + (n+1)^2 = 85 \).

["# Solving the Equation: ( n^2 + (n+1)^2 = 85 )", "When faced with the equation ( n^2 + (n+1)^2 = 85 ), many learners want to understand how to simplify and solve it efficiently. This seemingly simple quadratic equation appears frequently in algebra, geometry, and number-based problems. In this article, we explore step-by-step how to solve this equation, discover its solutions, and understand the underlying mathematical principles. Whether you're a student tackling homework or a curious learner exploring algebra, this guide is your go-to resource.", "## Step-by-Step Solution", "The equation is:", "[\nn^2 + (n+1)^2 = 85\n]", "Step 1: Expand the squared term", "Begin by expanding ( (n+1)^2 ):", "[\n(n+1)^2 = n^2 + 2n + 1\n]", "Substitute back into the equation:", "[\nn^2 + n^2 + 2n + 1 = 85\n]", "Step 2: Combine like terms", "Combine the ( n^2 ) terms:", "[\n2n^2 + 2n + 1 = 85\n]", "Step 3: Move all terms to one side", "Subtract 85 from both sides:", "[\n2n^2 + 2n + 1 - 85 = 0\n]", "[\n2n^2 + 2n - 84 = 0\n]", "Step 4: Simplify the quadratic equation", "Divide the entire equation by 2 to reduce coefficients:", "[\nn^2 + n - 42 = 0\n]", "Step 5: Solve the quadratic using factoring", "We seek two numbers whose product is ( -42 ) and sum is ( 1 ). These numbers are ( 7 ) and ( -6 ):", "[\n(n + 7)(n - 6) = 0\n]", "Step 6: Find the solutions", "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]", "## Interpreting the Solutions", "The equation ( n^2 + (n+1)^2 = 85 ) yields two real solutions:", "- ( n = -7 )\n- ( n = 6 )", "💡 Both values satisfy the original equation:", "- For ( n = 6 ):\n ( 6^2 + 7^2 = 36 + 49 = 85 ) ✅\n- For ( n = -7 ):\n ( (-7)^2 + (-6)^2 = 49 + 36 = 85 ) ✅", "These integer solutions make the equation particularly useful for teaching pattern recognition, expansion, and quadratic solving techniques.", "## Why This Equation Matters in Education and Math", "Equation form like ( n^2 + (n+1)^2 = 85 ) demonstrates fundamental concepts:", "- Algebraic manipulation: Expanding and simplifying expressions.\n- Quadratic equations: Recognizing the standard form ( ax^2 + bx + c = 0 ).\n- Verification: Testing integer solutions supports algebraic reasoning.\n- Application: Such expressions appear in real-world scenarios such as geometry (sum of areas of squares), number puzzles, and optimization problems.", "## Practice Problems to Master the Solution Pattern", "Looking to test your understanding? Try solving these variations:", "1. ( n^2 + (n+1)^2 = 65 )\n2. ( (n-2)^2 + (n+3)^2 = 130 )\n3. Find all integer solutions to ( n^2 + (n+1)^2 = k ) for general ( k )", "Each of these problems reinforces quadratic solving and pattern recognition.", "## Conclusion", "The equation ( n^2 + (n+1)^2 = 85 ) serves as a clear and accessible example of solving quadratic expressions through expansion, simplification, and factoring. With solutions ( n = -7 ) and ( n = 6 ), it illustrates how algebraic structures converge to elegant results. Understanding this equation improves both basic algebra skills and deeper mathematical reasoning—essential foundations for studying higher-level mathematics.", "Whether you're studying for tests, teaching students, or exploring math on your own, mastering equations like this strengthens numeracy confidence and problem-solving agility. Start by practicing simplification, then advance to applying these methods across various quadratic forms.", "---", "### Key Keywords for SEO: \nEquationSolve #n2Plus(n1Plus1)2Equals85 #Algebra #QuadraticEquations #SolveForN #MathLearning #AlgebraicExpression #EducationalMath #QuadraticPatterns", "---\nWant more step-by-step algebra guides? Subscribe to our math series and unlock hundreds of solved equations and problem-solving tips!"]