The equation is \( n^2 + (n + 1)^2 = 85 \).

["Solve: ( n^2 + (n + 1)^2 = 85 ) – A Step-by-Step Equation Guide", "Mathematics often hides elegant solutions within seemingly simple equations. One such equation that sparks curiosity and problem-solving is:", "[\nn^2 + (n + 1)^2 = 85\n]", "This article walks you through solving this quadratic equation step-by-step, helping you understand not only how to find the value of ( n ), but also why this process matters. Whether you're a student tackling algebra for the first time or a math enthusiast brushing up on fundamentals, solving this equation is a perfect exercise.", "---", "### Understanding the Equation", "Start with the original equation:", "[\nn^2 + (n + 1)^2 = 85\n]", "Here, ( n^2 ) represents the square of ( n ), and ( (n + 1)^2 ) is the square of the next consecutive integer. The sum of these two squares equals 85—a small integer, making this a manageable quadratic problem.", "---", "### Step-by-Step Solution", "#### Step 1: Expand the expression\nExpand ( (n + 1)^2 ):", "[\n(n + 1)^2 = n^2 + 2n + 1\n]", "Substitute into the original equation:", "[\nn^2 + (n^2 + 2n + 1) = 85\n]", "#### Step 2: Combine like terms\nCombine the ( n^2 ) terms and simplify:", "[\n2n^2 + 2n + 1 = 85\n]", "#### Step 3: Move all terms to one side\nSubtract 85 from both sides:", "[\n2n^2 + 2n + 1 - 85 = 0\n]\n[\n2n^2 + 2n - 84 = 0\n]", "#### Step 4: Simplify the quadratic equation\nDivide the entire equation by 2 to reduce coefficients:", "[\nn^2 + n - 42 = 0\n]", "#### Step 5: Solve the simplified quadratic\nNow solve using factoring, completing the square, or the quadratic formula. This equation factors neatly:", "[\nn^2 + n - 42 = (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]", "---", "### Verifying the Solutions", "Plugging ( n = 6 ) back in:", "[\n6^2 + (6 + 1)^2 = 36 + 49 = 85 \quad \ ext{✔}\n]", "Plugging ( n = -7 ):", "[\n(-7)^2 + (-7 + 1)^2 = 49 + 36 = 85 \quad \ ext{✔}\n]", "Both values satisfy the equation—meaning both are valid real solutions.", "---", "### Why This Equation Matters", "Solving ( n^2 + (n + 1)^2 = 85 ) illustrates key algebraic concepts: expanding binomials, combining terms, simplifying quadratics, and applying the zero-product property. It demonstrates how real-world problems can be modeled mathematically and manipulated to uncover precise answers.", "Moreover, consecutive squares appearing together are common in number theory and contest math, making this equation a gateway to deeper mathematical exploration.", "---", "### Final Thoughts", "The equation ( n^2 + (n + 1)^2 = 85 ) might look simple, but it’s a perfect example of how mathematical thinking balances intuition with systematic steps. Whether you're solving exams, learning algebra, or solving real-life puzzles, mastering such equations strengthens analytical skills and confidence.", "Try solving it now:\nFind all real values of ( n ) satisfying ( n^2 + (n + 1)^2 = 85 )—you already know the answer is ( n = 6 ) or ( n = -7 ), and understanding how leads to lasting knowledge.", "---", "Keywords: \nSolveQuadraticEquation #nSquaredPlus(nPlus1)Squared = 85 #AlgebraExercise #EquationSolving #ConsecutiveIntegers #QuadraticRoots #MathTips #EducationalContent", "Meta Description:\nLearn how to solve ( n^2 + (n + 1)^2 = 85 )—step-by-step. Understand factoring, simplification, and verify solutions efficiently. Perfect for students and math learners.", "---", "If you found this guide helpful, share it with your peers. Algebra wins when shared."]