En développant, on obtient : \( x^2 + x^2 + 2x + 1 = 85 \).

["Title: Solving the Quadratic Equation That Emerges in Algebra: (x^2 + x^2 + 2x + 1 = 85)", "---", "Introduction\nIn algebra, simplifying expressions often leads to elegant quadratic equations that reveal deeper mathematical insights. One such expression appears when developing certain identities or completing the square:", "[\nx^2 + x^2 + 2x + 1 = 85\n]", "This equation may seem straightforward at first, but solving it demonstrates core algebraic techniques and offers practical applications in modeling and optimization. This article walks you through simplifying and solving this equation, highlighting key steps and revealing the quadratic form it represents.", "---", "Step 1: Simplify the Expression\nBegin by combining like terms on the left-hand side:\n[\nx^2 + x^2 + 2x + 1 = 2x^2 + 2x + 1\n]", "So the equation becomes:\n[\n2x^2 + 2x + 1 = 85\n]", "---", "Step 2: Bring All Terms to One Side\nSubtract 85 from both sides to form a standard quadratic equation:\n[\n2x^2 + 2x + 1 - 85 = 0 \implies 2x^2 + 2x - 84 = 0\n]", "Divide the entire equation by 2 to simplify further:\n[\nx^2 + x - 42 = 0\n]", "---", "Step 3: Solve the Quadratic Equation\nNow we have a clean quadratic equation:\n[\nx^2 + x - 42 = 0\n]", "This can be solved using the quadratic formula:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "For (a = 1), (b = 1), and (c = -42):\n[\nx = \frac{-1 \pm \sqrt{1^2 - 4(1)(-42)}}{2(1)} = \frac{-1 \pm \sqrt{1 + 168}}{2} = \frac{-1 \pm \sqrt{169}}{2}\n]\n[\nx = \frac{-1 \pm 13}{2}\n]", "This gives two solutions:\n[\nx = \frac{-1 + 13}{2} = \frac{12}{2} = 6 \quad \ ext{and} \quad x = \frac{-1 - 13}{2} = \frac{-14}{2} = -7\n]", "---", "Step 4: Verify and Interpret the Solutions\nBoth (x = 6) and (x = -7) satisfy the original equation. This shows the equation has two real roots, typical for a properly formed quadratic.", "Applications of solving such equations include physics modeling (e.g., projectile motion), economics (profit maximization), and computer science (optimization algorithms). Recognizing these patterns helps in faster problem-solving across disciplines.", "---", "Conclusion\nThe seemingly simple equation (x^2 + x^2 + 2x + 1 = 85) transforms seamlessly into a standard quadratic (x^2 + x - 42 = 0), solvable via classic algebraic methods. Mastering these steps builds a foundation for tackling more complex polynomials and understanding real-world functional relationships. Whether studying for exams or applying concepts in practical fields, grasping how to simplify and solve quadratic equations is invaluable.", "Continue exploring related topics—like completing the square or analyzing discriminants—to deepen your algebraic fluency.", "---", "Keywords: quadratic equation, solving (x^2 + x - 42 = 0), algebra simplification, quadratic formula, real roots, solving equations, step-by-step math, math tutorial, algebra practice", "---", "Want more algebraic insights? Check out our guides on factoring quadratics, interpreting graphs, and real-world equation applications."]