\frac{3}{2}s^2 = 54
["### Understanding the Equation: \frac{3}{2}s² = 54 – A Step-by-Step Guide", "Mathematics is often best understood through real-world problems and clear solutions. One such equation that arises frequently in algebra, geometry, and physics is:", "$$\n\frac{3}{2}s^2 = 54\n$$", "Whether you're a student learning quadratic equations, a teacher explaining algebraic manipulation, or a curious learner solving practical problems, this equation offers a great example of solving for an unknown variable.", "---", "### Step 1: Isolate the Variable Term", "To solve for ( s ), begin by isolating ( s^2 ). Since ( s^2 ) is multiplied by ( \frac{3}{2} ), divide both sides of the equation by ( \frac{3}{2} ):", "$$\ns^2 = \frac{54}{\frac{3}{2}}\n$$", "Dividing by a fraction is the same as multiplying by its reciprocal:", "$$\ns^2 = 54 \ imes \frac{2}{3} = 36\n$$", "---", "### Step 2: Solve for ( s )", "Now, take the square root of both sides:", "$$\ns = \pm\sqrt{36} = \pm 6\n$$", "---", "### Final Answer", "The solutions to the equation \frac{3}{2}s² = 54 are:", "$$\ns = 6 \quad \ ext{or} \quad s = -6\n$$", "---", "### Why This Equation Matters", "Understanding how to solve quadratic-like equations is essential in many fields:", "- Algebra and Pre-Calculus: Building foundational skills for graphing parabolas, solving quadratic equations, and working with squares.\n- Physics: Modeling motion, energy, and wave behavior often involves quadratic relationships.\n- Engineering & Economics: Many real-world systems exhibit proportional or squared dependencies, requiring algebraic solutions.", "---", "### How to Verify the Solution", "Plug ( s = 6 ) back into the original equation:", "$$\n\frac{3}{2}(6)^2 = \frac{3}{2} \cdot 36 = 54 \quad \ ext{✔}\n$$", "Similarly, for ( s = -6 ):", "$$\n\frac{3}{2}(-6)^2 = \frac{3}{2} \cdot 36 = 54 \quad \ ext{✔}\n$$", "Both values satisfy the equation.", "---", "### Summary", "The equation ( \frac{3}{2}s^2 = 54 ) simplifies neatly to ( s^2 = 36 ), leading to two real solutions: ( +\sqrt{36} = 6 ) and ( -\sqrt{36} = -6 ). Mastering this step-by-step approach not only solves the immediate problem but strengthens problem-solving skills applicable across various disciplines.", "If you found this explanation helpful, consider exploring related topics like solving linear and quadratic inequalities, or using algebraic identities to simplify expressions.", "---", "Keywords: solve (\frac{3}{2}s^2 = 54), quadratic equations, algebraic manipulation, isolate variable, step-by-step solution, math tutorial, algebra help."]
