\frac{3\sqrt{3}}{2}s^2 = 54\sqrt{3}

["# Solve the Equation: $\frac{3\sqrt{3}}{2}s^2 = 54\sqrt{3}$", "Understanding how to solve equations involving radicals and coefficients is essential in algebra, especially when tackling geometry, physics, or engineering problems. One classic example is solving equations like\n$$\n\frac{3\sqrt{3}}{2}s^2 = 54\sqrt{3}.\n$$\nIn this article, we’ll break down step-by-step how to isolate $ s^2 $ and find the value of $ s $, explaining key concepts along the way.", "## Step 1: Eliminate the coefficient on $ s^2 $", "We begin with:\n$$\n\frac{3\sqrt{3}}{2}s^2 = 54\sqrt{3}.\n$$\nTo isolate $ s^2 $, divide both sides of the equation by $ \frac{3\sqrt{3}}{2} $. Remember, dividing by a fraction is the same as multiplying by its reciprocal:\n$$\ns^2 = 54\sqrt{3} \div \left( \frac{3\sqrt{3}}{2} \right) = 54\sqrt{3} \cdot \frac{2}{3\sqrt{3}}.\n$$", "## Step 2: Simplify the expression", "Cancel $ \sqrt{3} $ terms (since they appear in numerator and denominator):\n$$\ns^2 = 54 \cdot \frac{2}{3} = \frac{108}{3} = 36.\n$$", "## Step 3: Solve for $ s $", "Now that $ s^2 = 36 $, take the square root of both sides:\n$$\ns = \pm \sqrt{36} = \pm 6.\n$$", "Since $ s $ typically represents a length or physical quantity, only the positive solution is often valid unless context specifies otherwise.", "## Why This Equation Matters", "Equations of this form frequently arise when calculating areas or scaling in geometric problems, particularly involving equilateral triangles, because $ \sqrt{3} $ commonly appears in trigonometric expressions related to 60° angles. Solving for $ s^2 $ efficiently is crucial in optimization and applied math settings.", "## Final Answer", "$$\n\boxed{s = \pm 6}\n$$", "or, if considering only the positive physical value,\n$$\n\boxed{s = 6}.\n$$", "---", "### SEO Meta Overview:", "- Main Keyword: $\frac{3\sqrt{3}}{2}s^2 = 54\sqrt{3}$\n- Long-Tail Keywords: Solving radical equations, algebraic simplification, square root solving, geometry applications, solving for $s^2$, mathematical steps explained\n- TParents: Algebra, high school math, equation solving, radical equations, geometry problems, calculus prep\n- Focus: Clear step-by-step solving, simplification with radicals, real-world relevance.", "---", "Optimized heading:\nHow to Solve $\frac{3\sqrt{3}}{2}s^2 = 54\sqrt{3}$: Step-by-Step Guide\nSimplify and Find $s$ in Radical Equations"]