3n² + 2 = 425 → 3n² = 423 → n² = 141 → n = √141 ≈ 11.87 → not integer.
["Understanding the Quadratic Equation: 3n² + 2 = 425 and Why Its Solution Isn’t an Integer", "When solving quadratic equations, one of the most common steps is isolating the variable term to make further analysis possible. A real-world example that illustrates this concept is the equation:\n3n² + 2 = 425", "This straightforward equation serves as an excellent starting point to explore the principles of algebraic manipulation and the nature of solutions—especially when the final answer is not an integer.", "---", "### Step-by-Step Solution Breakdown", "We begin with the original equation:", "[ 3n² + 2 = 425 ]", "Step 1: Isolate the quadratic term\nSubtract 2 from both sides to eliminate the constant on the left:\n[ 3n² = 425 - 2 ]\n[ 3n² = 423 ]", "Step 2: Solve for ( n² )\nDivide both sides by 3:\n[ n² = \frac{423}{3} = 141 ]", "Step 3: Take the square root\nNow, take the square root of both sides to solve for ( n ):\n[ n = \sqrt{141} ]", "Since ( 141 ) is not a perfect square,\n[ n = \sqrt{141} \approx 11.874 ]", "---", "### Why Isn’t n an Integer?", "An important lesson here is that not all quadratic equations yield integer solutions. In this case, because ( n² = 141 ) and √141 is irrational (approximately 11.874), ( n ) cannot be expressed as a whole number. This result arises naturally from the algebra: ( 141 ) lies between the perfect squares 121 ((11^2)) and 169 ((13^2)), confirming (n) is not an integer.", "---", "### Practical Implications and Applications", "While non-integer solutions in algebra might seem abstract, they frequently appear in real-world modeling—especially in physics, engineering, and computer science—where exact values assure precision in measurements, design, or computational efficiency. Understanding why ( n = \sqrt{141} ) is irrational helps reinforce number theory concepts and supports accurate decision-making in technical applications.", "---", "### Summary", "- Start with: ( 3n² + 2 = 425 )\n- Isolate ( n² ): ( n² = 141 )\n- Solve for ( n ): ( n = \sqrt{141} \approx 11.87 )\n- Recognize that ( n ) is not an integer because 141 is not a perfect square", "This simple quadratic exemplifies how algebraic methods expose meaningful patterns—even when exact answers are irrational. Whether you’re a student, educator, or professional, mastering such equations builds a strong foundation in problem-solving and mathematical reasoning.", "---", "Want to explore more? Practice similar equations like ( 2n² - 6 = 10 ) or ( 4n² + 16 = 288 )—each reveals unique insights about roots, domains, and number types. Download our free quadratic equation practice worksheet and sharpen your algebra skills today!"]
