L'équation est : \( x^2 + (x+1)^2 = 85 \).

["# Solving L'Équation : ( x^2 + (x + 1)^2 = 85 ) — A Step-by-Step Guide", "If you're exploring quadratic equations this season, few problems are as engaging and accessible as solving\n( x^2 + (x + 1)^2 = 85 ). This elegant equation combines algebra and geometry, offering both challenge and insight into the beauty of mathematical structure. In this SEO-optimized article, we will guide you through solving this equation step by step, explain the underlying concepts, boost your keyword rankings, and help learners—whether students, educators, or math enthusiasts—understand and master the process.", "---", "## What is the Equation ( x^2 + (x + 1)^2 = 85 )?", "At first glance, the expression ( x^2 + (x + 1)^2 ) might seem cryptic, but expanding it reveals a simple quadratic:\n[\nx^2 + (x + 1)^2 = x^2 + (x^2 + 2x + 1) = 2x^2 + 2x + 1\n]\nSo the equation becomes:\n[\n2x^2 + 2x + 1 = 85\n]", "This simplifies to:\n[\n2x^2 + 2x - 84 = 0\n]\nFurther divide all terms by 2:\n[\nx^2 + x - 42 = 0\n]", "Now that we’ve transformed the original equation into a standard quadratic in its simplest form, solving it becomes straightforward using familiar techniques like factoring, the quadratic formula, or completing the square.", "---", "## Step-by-Step Solution", "### Step 1: Expand and simplify\nStart with:\n[\nx^2 + (x + 1)^2 = 85\n]\nExpand:\n[\nx^2 + x^2 + 2x + 1 = 85\n]\nCombine like terms:\n[\n2x^2 + 2x + 1 = 85\n]\nSubtract 85 from both sides:\n[\n2x^2 + 2x - 84 = 0\n]\nDivide the entire equation by 2:\n[\nx^2 + x - 42 = 0\n]", "### Step 2: Solve using factoring\nLook for two numbers that multiply to (-42) and add to (+1). These numbers are (7) and (-6), since:\n[\n7 \ imes (-6) = -42\quad \ ext{and} \quad 7 + (-6) = 1\n]\nSo, factor the quadratic:\n[\n(x + 7)(x - 6) = 0\n]", "### Step 3: Find the roots\nSet each factor equal to zero:\n[\nx + 7 = 0 \implies x = -7\n]\n[\nx - 6 = 0 \implies x = 6\n]", "---", "## Answer and Verification", "The solutions are:\n[\n\boxed{x = -7 \quad \ ext{and} \quad x = 6}\n]", "Check these values in the original equation:", "- For ( x = 6 ):\n[\n6^2 + (6+1)^2 = 36 + 49 = 85 \quad \checkmark\n]", "- For ( x = -7 ):\n[\n(-7)^2 + (-7 + 1)^2 = 49 + (-6)^2 = 49 + 36 = 85 \quad \checkmark\n]", "Both solutions satisfy the equation — confirming correctness.", "---", "## Math Concepts and Educational Value", "This equation beautifully illustrates:", "- Algebraic expansion and simplification — combining terms and removing parentheses.\n- Quadratic equation setup — transforming expressions into standard form ( ax^2 + bx + c = 0 ).\n- Double root variation — the expression ( x^2 + (x+1)^2 ) naturally produces a quadratic due to the linear shift.\n- Symmetry in roots — despite algebraic complexity, roots emerge symmetrically around a central value.", "---", "## Why This Equation Matters for SEO and Learning", "1. High-traffic keywords: Phrases like "solve ( x^2 + (x + 1)^2 = 85 )", "quadratic equation solutions", and "how to solve ( x^2 + (x+1)^2 = 85 " attract middle and high school students, educators, and math learners worldwide.", "2. Beginner-friendly yet insightful: It’s simple enough for beginners but deep enough to reinforce quadratic fundamentals — perfect for blog posts targeting math students or educators seeking fresh content.", "3. Engagement boost: The step-by-step video or article format enhances dwell time, audio-visual SEO signals, and shareability on educational platforms.", "4. Link-building opportunity: This equation is a natural hub linking to related articles: quadratic formulas, graphing parabolas, completing the square, and real-world applications.", "---", "## Conclusion", "The equation\n[\nx^2 + (x + 1)^2 = 85\n]\nserves as a gateway to essential algebraic techniques. By solving it methodically — from expansion to factoring — learners gain confidence and mastery. Whether you’re a student, teacher, or content creator, leveraging this equation enriches both understanding and online visibility.", "---", "Keywords for SEO optimization: \nSolve (x^2 + (x+1)^2 = 85), #QuadraticEquations, #AlgebraSolutions, #SolveQuadraticEquation, #MathTutorial, #LearnAlgebra", "---", "Meta Title:\nSolve ( x^2 + (x + 1)^2 = 85 ) — Step-by-Step Guide with Solutions and Explanations", "Meta Description:\nLearn how to solve the equation ( x^2 + (x + 1)^2 = 85 ) with clear steps, simplification techniques, and verified solutions. Perfect for students and math learners.", "---", "Transform this equation from a divisional problem into a powerful teaching and SEO asset — discover the elegance of quadratic solutions today!"]