x^2 \left( \frac{64 - 100}{6400} \right) = 0 \Rightarrow x^2 \cdot (-36/6400) = 0
["Solving the Equation: ( x^2 \cdot \left( \frac{64 - 100}{6400} \right) = 0 \Rightarrow x^2 \cdot \left( -\frac{36}{6400} \right) = 0 )", "Understanding how to solve quadratic-style equations involving proportional expressions is essential in algebra. This article breaks down the equation ( x^2 \cdot \left( \frac{64 - 100}{6400} \right) = 0 ) and explains why it simplifies cleanly to ( x^2 \cdot \left( -\frac{36}{6400} \right) = 0 ), leading to the solution — and why there are no real solutions.", "---", "### Step 1: Simplify the Expression Inside the Parentheses", "Start by evaluating the numerator ( 64 - 100 ):", "[\n64 - 100 = -36\n]", "So the original equation becomes:", "[\nx^2 \cdot \left( \frac{-36}{6400} \right) = 0\n]", "This simplifies to:", "[\nx^2 \cdot \left( -\frac{36}{6400} \right) = 0\n]", "---", "### Step 2: Understand the Equation Form", "The expression ( x^2 \cdot \left( -\frac{36}{6400} \right) = 0 ) is a constant multiple equation: a squared term multiplied by a nonzero constant equals zero.", "Recall the zero product property:\nIf the product of two factors is zero, then at least one of the factors must be zero.", "Here, the factor ( x^2 ) and the constant ( -\frac{36}{6400} ) are multiplied. Since ( -\frac{36}{6400} <br/>\ne 0 ), the only way the product is zero is if:", "[\nx^2 = 0\n]", "---", "### Step 3: Solve for ( x )", "Solve ( x^2 = 0 ):", "[\nx = 0\n]", "No other real solution exists because a square ( x^2 ) is always nonnegative, and equals zero only when ( x = 0 ).", "---", "### Step 4: Why the Equation Holds", "The original equation", "[\nx^2 \cdot \left( \frac{-36}{6400} \right) = 0\n]", "forces ( x^2 ) to balance the nonzero negative constant. Only when ( x^2 = 0 ) does the product become exactly zero. There is no real ( x ) that makes a negative number zero — only ( x = 0 ) nullifies ( x^2 ).", "---", "### Step 5: Final Verification", "Plug ( x = 0 ) back into the original expression:", "[\nx^2 \cdot \left( \frac{64 - 100}{6400} \right) = (0)^2 \cdot \left( \frac{-36}{6400} \right) = 0\n]", "True.", "---", "### Summary", "- The equation simplifies from ( x^2 \cdot \left( \frac{-36}{6400} \right) = 0 ) directly via the zero product property.\n- Since ( -\frac{36}{6400} <br/>\ne 0 ), the only solution is ( x^2 = 0 ).\n- Therefore, ( x = 0 ) is the only real solution to the equation.\n- This problem illustrates how constant coefficients combined with ( x^2 ) allow clean factorization and precise solution determination.", "---", "### Bonus Tips for Solving Similar Equations", "- Always simplify expressions inside parentheses first.\n- Apply the zero product property carefully when dealing with products involving constants.\n- Recognize that ( x^2 = 0 ) has one real solution: ( x = 0 ).\n- Use simplification to isolate the squared term when the constant factor is nonzero.", "---", "Keywords:\n( x^2 \cdot \left( \frac{64 - 100}{6400} \right) = 0 ), ( x^2 \cdot \left( -\frac{36}{6400} \right) = 0 ), how to solve quadratic equations, algebra factored equations, zero product property, solving equations with constants, real solutions, ( x = 0 )", "---", "Related Read: Learn how to isolate variables in proportional equations and apply the zero product property effectively to solve algebraic expressions quickly and accurately."]
