\[ (n^2 - 2n + 1) + n^2 + (n^2 + 2n + 1) = 425 \]

["Solve the Equation: ( (n^2 - 2n + 1) + n^2 + (n^2 + 2n + 1) = 425 )", "Understanding and solving quadratic equations is a foundational skill in algebra, essential for students, educators, and math enthusiasts. Today, we break down and solve the equation:", "[\n(n^2 - 2n + 1) + n^2 + (n^2 + 2n + 1) = 425\n]", "This equation resembles a common pattern in quadratic expressions, offering insight into simplifying algebraic expressions and solving real-world problems.", "---", "### Step 1: Simplify the Left-Hand Side", "Start by combining like terms on the left-hand side.", "[\n(n^2 - 2n + 1) + n^2 + (n^2 + 2n + 1)\n]", "Group the (n^2), (n), and constant terms:", "- (n^2 + n^2 + n^2 = 3n^2)\n- (-2n + 2n = 0) (the linear terms cancel out)\n- (1 + 1 = 2) (constant terms add to 2)", "Thus, the expression simplifies to:", "[\n3n^2 + 2\n]", "---", "### Step 2: Set the Equation Equal to 425", "Now rewrite the original equation as:", "[\n3n^2 + 2 = 425\n]", "Subtract 2 from both sides:", "[\n3n^2 = 423\n]", "---", "### Step 3: Solve for (n^2)", "Divide both sides by 3:", "[\nn^2 = \frac{423}{3} = 141\n]", "---", "### Step 4: Solve for (n)", "Take the square root of both sides:", "[\nn = \pm \sqrt{141}\n]", "Since (141) is not a perfect square, the solutions are irrational. However, both values are real numbers satisfying the equation.", "---", "### Why This Equation Matters", "Solving expressions like this builds algebraic fluency and provides practice in simplification, combining like terms, and manipulating equations. This type of quadratic simplification often appears in physics, engineering, and economics when modeling relationships or optimizing functions.", "---", "### Final Answer", "The solutions to the equation\n[\n(n^2 - 2n + 1) + n^2 + (n^2 + 2n + 1) = 425\n]\nare:", "[\nn = \sqrt{141} \quad \ ext{and} \quad n = -\sqrt{141}\n]", "---", "Keywords:\nalgebra, quadratic equation, solve for n, simplify expression, solve linear + quadratic equations, 3n² + 2 = 425, step-by-step algebra, math solution, irrational numbers, algebraic simplification", "Meta Description:\nLearn how to solve ( (n^2 - 2n + 1) + n^2 + (n^2 + 2n + 1) = 425 ) step-by-step. Discover simplification techniques, combining like terms, and real-world applications of quadratic expressions."]