\[ 3n^2 + 2 = 425 \Rightarrow 3n^2 = 423 \Rightarrow n^2 = 141 \Rightarrow n = \sqrt{141} \]

["# Solving the Equation: ( 3n^2 + 2 = 425 ) Step-by-Step", "Understanding how to solve quadratic equations is essential for students, mathematicians, and anyone working with mathematical modeling. In this article, we’ll walk through solving the equation:", "[\n3n^2 + 2 = 425\n]", "Step-by-step breakdown makes complex problems easier to grasp—and this equation is a perfect example for learning foundational algebra skills.", "---", "## Step 1: Isolate the Quadratic Term", "Start by simplifying the equation to isolate the ( n^2 ) term:", "[\n3n^2 + 2 = 425\n]", "Subtract 2 from both sides:", "[\n3n^2 = 423\n]", "This step eliminates the constant term, making it easier to solve for ( n^2 ).", "---", "## Step 2: Solve for ( n^2 )", "Next, divide both sides by 3 to solve for ( n^2 ):", "[\nn^2 = \frac{423}{3} = 141\n]", "Dividing by 3 streamlines the equation and reveals the expression under the square root in the final step.", "---", "## Step 3: Take the Square Root", "To find ( n ), take the square root of both sides. Since squaring a number yields both positive and negative results, we include both:", "[\nn = \pm \sqrt{141}\n]", "Note: ( \sqrt{141} ) is an irrational number—approximately 11.87—but it represents the exact value.", "---", "## Why This Equation Matters", "This seemingly simple equation introduces learners to solving quadratic forms by isolating variables and using inverse operations. Mastery of this pattern prepares you for more complex polynomial equations and real-world applications involving area, motion, and optimization.", "---", "## Final Answer", "[\nn = \pm \sqrt{141}\n]", "---", "## Additional Tips for Solving Similar Equations", "- Always isolate the quadratic term first.\n- Perform operations symmetrically on both sides to maintain equality.\n- When roots are irrational, leave them in radical form unless a decimal approximation is required.\n- Practice factoring, completing the square, and using the quadratic formula to expand your tools.", "---", "### Keywords for SEO:\nsolve 3n² + 2 = 425, quadratic equation steps, how to solve n² = 141, alpha math tutorial, step-by-step algebra solution, n = sqrt(141)", "Understanding these fundamentals builds confidence and clarity in algebra—key stepping stones to advanced mathematics."]