x^2 \left( \frac{64 - 100}{6400} \right) = 0 \Rightarrow x^2(-36/6400) = 0

["# Solving the Equation: ( x^2 \left( \frac{64 - 100}{6400} \right) = 0 \Rightarrow x^2 \left( -\frac{36}{6400} \right) = 0 )", "Mathematics often presents equations that may seem simple at first glance but carry deeper insights into algebra and solutions. One such concise equation is:", "[\nx^2 \left( \frac{64 - 100}{6400} \right) = 0\n]", "Simplifying the expression, we begin by evaluating the coefficient:", "[\n\frac{64 - 100}{6400} = \frac{-36}{6400}\n]", "So the equation becomes:", "[\nx^2 \cdot \left( -\frac{36}{6400} \right) = 0\n]", "## Understanding the Equation", "This equation asks under what values of ( x ) the product ( x^2 \cdot \left( -\frac{36}{6400} \right) ) equals zero. Recall a fundamental rule of algebra: a product of factors equals zero if and only if at least one of the factors is zero. Since ( -\frac{36}{6400} ) is a non-zero constant, the entire expression is zero only when:", "[\nx^2 = 0\n]", "### Solving for ( x )", "To solve ( x^2 = 0 ), take the square root of both sides:", "[\nx = 0\n]", "This is the only solution to the equation. The presence of ( x^2 ) ensures that only ( x = 0 ) makes the entire expression zero, regardless of the constant fraction.", "## Why This Matters in Algebra", "This problem showcases how equations involving squares and multiplicative constants reduce to simple root-finding. It also illustrates:", "- The importance of coefficients: the fraction (-\frac{36}{6400}) is non-zero but doesn’t affect the solution when multiplied by ( x^2 ).\n- The property of zero products: only one factor needs to be zero to make the overall expression zero.\n- How rational expressions (ratios like (-36/6400)) behave when multiplied by polynomials like (x^2).", "## Real-World Context", "Such equations appear in physics and engineering when modeling relationships involving quadratic expressions under constraints — for example, in optimization problems or when analyzing forces in equilibrium where a squared term vanishes under specific input values.", "## Summary", "The equation\n[\nx^2 \left( \frac{64 - 100}{6400} \right) = 0\n]\nsimplifies elegantly to\n[\nx^2 \left( -\frac{36}{6400} \right) = 0\n]\nand conclusively yields:\n[\nx = 0\n]\nThis single solution arises from the algebraic principle that zero multiplied by a non-zero number equals zero — and only ( x = 0 ) satisfies that condition.", "---", "Keywords: solve x²(64 - 100)/6400 = 0, x² × (-36/6400) = 0, algebraic equation solution, x = 0, zero product property, square equation, math problem solving", "Location: Optimize your algebra skills by mastering how constant factors influence roots. Understand why (x = 0) is the only solution in this case — a clear, Students, educators, and math enthusiasts will benefit from this concise explanation."]