Given $ \frac{\sqrt{3}}{4} s^2 = 36\sqrt{3} $, divide both sides by $ \sqrt{3} $:
["Solving Quadratic Equations Step-by-Step: Given $ \frac{\sqrt{3}}{4} s^2 = 36\sqrt{3} $", "When solving equations involving variables with square roots, algebraic manipulation is essential. One common step in simplifying such equations is dividing both sides by $ \sqrt{3} $. In this article, we’ll explore how this operation helps solve the equation:", "$$\n\frac{\sqrt{3}}{4} s^2 = 36\sqrt{3}\n$$", "### Step 1: Divide Both Sides by $ \sqrt{3} $", "Starting with the given equation:", "$$\n\frac{\sqrt{3}}{4} s^2 = 36\sqrt{3}\n$$", "Divide both sides by $ \sqrt{3} $:", "$$\n\frac{ \frac{\sqrt{3}}{4} s^2 }{ \sqrt{3} } = \frac{36\sqrt{3}}{\sqrt{3}}\n$$", "Simplify both sides:", "- Left side:\n$$\n\frac{\sqrt{3}}{\sqrt{3}} \cdot \frac{s^2}{4} = \frac{s^2}{4}\n$$\n- Right side:\n$$\n\frac{36\sqrt{3}}{\sqrt{3}} = 36\n$$", "Now the simplified equation becomes:", "$$\n\frac{s^2}{4} = 36\n$$", "### Step 2: Solve for $ s^2 $", "Multiply both sides by 4 to isolate $ s^2 $:", "$$\ns^2 = 36 \ imes 4 = 144\n$$", "### Step 3: Solve for $ s $", "Take the square root of both sides:", "$$\ns = \pm \sqrt{144} = \pm 12\n$$", "### Final Answer", "The solutions to the equation $ \frac{\sqrt{3}}{4} s^2 = 36\sqrt{3} $ are:", "$$\ns = 12 \quad \ ext{or} \quad s = -12\n$$", "### Why This Step Matters", "Dividing by $ \sqrt{3} $ simplifies the original equation by removing the radical from the coefficient of $ s^2 $, making the equation easier to solve. This technique is particularly useful in algebra and calculus when isolating variables in quadratic expressions involving radicals.", "---", "Key Takeaways:\n- Dividing both sides of an equation by the same nonzero value preserves equality.\n- Eliminating radicals from coefficients simplifies solving.\n- Always check your solution by substituting back into the original equation.", "Mastering such algebraic steps strengthens problem-solving skills for solving quadratic and radical equations, commonly encountered in science, engineering, and advanced mathematics.", "---", "Optimizing for SEO:", "This article targets web searches related to solving $ \frac{\sqrt{3}}{4} s^2 = 36\sqrt{3} $, dividing by $ \sqrt{3} $, and step-by-step algebra. Adding keywords like “solve quadratic equation with radicals,” “algebraic manipulation steps,” and “how to divide both sides by $ \sqrt{3} $” enhances SEO relevance for students, educators, and learners focusing on mathematics fundamentals."]
