\[ 3n^2 = 423 \] - United Radiology

["# Solving the Equation 3n² = 423: A Step-by-Step Guide", "Understanding how to solve quadratic equations is a fundamental skill in algebra. One common equation students encounter is 3n² = 423, which appears simple but opens the door to key problem-solving techniques. In this article, we’ll break down how to solve 3n² = 423, explain the underlying concepts, and explore practical applications of this type of equation.", "---", "## What Is the Equation 3n² = 423?", "The equation 3n² = 423 is a quadratic equation in standard form, though it's already simplified. It asks: What value(s) of n makes three times the square of n equal 423? Solving it helps reinforce algebraic manipulation, factoring, and understanding of square roots.", "---", "## Step-by-Step Solution", "To solve 3n² = 423, follow these straightforward algebraic steps:", "### Step 1: Isolate n²
\nDivide both sides of the equation by 3:
\n[
\n\frac{3n^2}{3} = \frac{423}{3}
\n]
\n[
\nn^2 = 141
\n]", "### Step 2: Solve for n by Taking the Square Root
\nApply the square root property: if ( n^2 = 141 ), then
\n[
\nn = \pm\sqrt{141}
\n]", "Since 141 is not a perfect square, the solution remains in radical form.", "### Final Answer:
\n[
\nn = \pm \sqrt{141}
\n]", "---", "## Why This Equation Matters", "### 1. Foundation for Quadratic Concepts
\nMastering equations like 3n² = 423 builds the foundation for solving more complex quadratics, including those with larger coefficients or x terms (e.g., ax² + bx + c = 0).", "### 2. Applications in Geometry
\nSuch equations model real-world scenarios:
\n- Finding dimensions of a square or rectangle with area constraints.
\n- Calculating side lengths when total area or perimeter is given in scaled form.", "### 3. Encouraging Algebraic Thinking
\nWorking with variables still in front of squared terms helps students develop precision in manipulating equations, a skill crucial in advanced math, physics, and engineering.", "---", "## How to Check the Solution", "To confirm our answer is correct, substitute ( n = \sqrt{141} ) and ( n = -\sqrt{141} ) back into the original equation:", "For ( n = \sqrt{141} ):
\n[
\n3(\sqrt{141})^2 = 3 \ imes 141 = 423
\n]
\nFor ( n = -\sqrt{141} ):
\n[
\n3(-\sqrt{141})^2 = 3 \ imes 141 = 423
\n]
\nBoth satisfy the equation, confirming our solution is complete.", "---", "## Alternative: Factoring and Rational Root Theorem", "While 3n² – 423 = 0 doesn’t factor nicely, understanding how to approach such equations prepares you for using the Quadratic Formula:
\n[
\nn = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\n]
\nIn this case, ( a = 3 ), ( b = 0 ), ( c = -423 ), so the formula simplifies to:
\n[
\nn = \pm \sqrt{141}
\n]", "---", "## Summary", "The equation 3n² = 423 may seem basic but serves as a gateway to deeper algebraic understanding. By isolating the squared term and applying square roots, you find:
\n[
\n\boxed{n = \pm \sqrt{141}}
\n]", "Mastering such equations strengthens problem-solving skills applicable in science, technology, and everyday math challenges. Keep practicing—each step brings you closer to math fluency!", "---", "## What’s Next?", "- Explore how to solve similar quadratic equations with linear terms
\n- Learn the difference between perfect square trinomials and standard quadratics
\n- Discover real-life problems involving squared variables", "Happy learning — keep solving!"]