\frac{x^2}{100} + \frac{64 - x^2}{64} = 1

["# Solving the Equation:\n$$\n\frac{x^2}{100} + \frac{64 - x^2}{64} = 1\n$$\nA Step-by-Step Guide with Key Insights", "---", "Introduction\nSolving algebraic equations like\n$$\n\frac{x^2}{100} + \frac{64 - x^2}{64} = 1\n$$\nmay seem challenging at first, but breaking it down step-by-step makes it manageable. This equation combines rational expressions with $ x^2 $, and solving it correctly unlocks insights into quadratic relationships and real-world applications such as modeling proportions and optimizing values.", "In this article, we’ll walk through how to simplify, solve, interpret, and apply this equation—ideal for students, educators, and anyone interested in mastering algebraic techniques.", "---", "## Step 1: Eliminate Denominators by Finding a Common Multiple", "The equation involves fractions with denominators 100 and 64. To eliminate these, multiply both sides of the equation by the least common multiple (LCM) of 100 and 64.", "The prime factorizations:\n- $100 = 2^2 \cdot 5^2$\n- $64 = 2^6$\nSo, LCM = $2^6 \cdot 5^2 = 64 \cdot 25 = 1600$", "Multiply every term by 1600:", "$$\n1600 \cdot \left( \frac{x^2}{100} \right) + 1600 \cdot \left( \frac{64 - x^2}{64} \right) = 1600 \cdot 1\n$$", "Simplify each term:\n- $ \frac{1600}{100} = 16 $ → $ 16x^2 $\n- $ \frac{1600}{64} = 25 $ → $ 25(64 - x^2) $\n- Right-hand side: $1600$", "Resulting equation:\n$$\n16x^2 + 25(64 - x^2) = 1600\n$$", "---", "## Step 2: Expand and Simplify", "Distribute 25 into $64 - x^2$:\n$$\n16x^2 + 25 \cdot 64 - 25x^2 = 1600\n$$\n$$\n16x^2 - 25x^2 + 1600 = 1600\n$$\n$$\n-9x^2 + 1600 = 1600\n$$", "---", "## Step 3: Isolate and Solve for $x^2$", "Subtract 1600 from both sides:\n$$\n-9x^2 = 0\n$$\n$$\nx^2 = 0\n$$", "---", "## Step 4: Final Solution", "Take the square root:\n$$\nx = 0\n$$", "---", "## Understanding the Solution", "The only real solution is $ x = 0 $. This makes sense algebraically: plugging $ x = 0 $ into the original equation gives:\n$$\n\frac{0^2}{100} + \frac{64 - 0^2}{64} = 0 + \frac{64}{64} = 1\n$$\nwhich satisfies the equation exactly.", "Graphically, this corresponds to the point where the function\n$$\nf(x) = \frac{x^2}{100} + \frac{64 - x^2}{64}\n$$\nintersects the horizontal line $ y = 1 $ — at $ x = 0 $ only.", "---", "## Why This Equation Matters", "- Modeling Proportions: The equation balances two weighted inputs against a target value — useful in physics, economics, and engineering for balancing ratios.\n- Quadratic Relationships: Although simplified to a linear form here, similar equations can model more complex curve intersections.\n- Verification of Simplifications: Understanding how denominator elimination works prevents errors in real-world mathematical modeling.", "---", "## Next Steps\nTo deepen your understanding:\n- Try solving related equations with different constants or variables.\n- Explore how changing coefficients affects the number and type of solutions (real, complex, repeated).\n- Consider real-world data that fits such proportional balances.", "---", "Conclusion\nThe equation\n$$\n\frac{x^2}{100} + \frac{64 - x^2}{64} = 1\n$$\nsimplifies cleanly to $ x = 0 $, offering a clear example of rational equation solving and quadratic structure. Mastering these steps strengthens algebraic skills essential for advanced math and applied sciences.", "---", "Keywords: Solve $ \frac{x^2}{100} + \frac{64 - x^2}{64} = 1 $, algebraic equation steps, quadratic simplification, solving rational equations, equation solving tutorial, real solutions, algebra practice problems", "---", "Explore more algebra articles to strengthen your problem-solving toolkit and unlock patterns behind complex equations."]