-\frac{5000}{x^2} + 0.5 = 0 \Rightarrow \frac{5000}{x^2} = 0.5 \Rightarrow x^2 = 10000 \Rightarrow x = 100

["# Solving the Equation: (-\frac{5000}{x^2} + 0.5 = 0) Step-by-Step", "Solving algebraic equations is a fundamental skill in mathematics, and mastering step-by-step methods ensures clear understanding and accurate results. Today, we’ll solve the equation:", "[\n-\frac{5000}{x^2} + 0.5 = 0\n]", "This equation may appear tricky at first, but by applying algebraic manipulation carefully, we can find the value of (x). Below is a clear and detailed breakdown of each step in solving ( -\frac{5000}{x^2} + 0.5 = 0 ).", "---", "## Step 1: Isolate the Fraction", "Begin by moving the constant term to the other side of the equation:", "[\n-\frac{5000}{x^2} + 0.5 = 0 \implies -\frac{5000}{x^2} = -0.5\n]", "Multiplying both sides by (-1) simplifies the signs:", "[\n\frac{5000}{x^2} = 0.5\n]", "---", "## Step 2: Eliminate the Denominator", "To eliminate the fraction, multiply both sides by (x^2):", "[\n5000 = 0.5 \cdot x^2\n]", "Now, divide both sides by 0.5 to isolate (x^2):", "[\nx^2 = \frac{5000}{0.5} = 10000\n]", "---", "## Step 3: Solve for (x)", "Now that we know (x^2 = 10000), take the square root of both sides:", "[\nx = \pm\sqrt{10000} = \pm 100\n]", "Thus, the solutions are (x = 100) and (x = -100).", "---", "## Final Answer", "[\n-\frac{5000}{x^2} + 0.5 = 0 \implies x = 100 \quad \ ext{or} \quad x = -100\n]", "---", "### Why This Equation Matters", "Equations like (-\frac{a}{x^2} + b = 0) appear in science and engineering, especially in modeling inverse-square relationships such as gravitational fields or signal decay. Solving them accurately ensures correct predictions and design valid models.", "---", "Key Takeaways:", "- Always isolate variables step by step.\n- Fractions can be cleared by multiplication.\n- Taking square roots introduces both positive and negative solutions.\n- Verification by plugging values back into the original equation confirms correctness.", "If you're learning algebra or need solutions with real-world applications, mastering this method simplifies more complex problems ahead.", "---", "Did you solve similar equations? Share your process in the comments!\nUnderstanding every step builds stronger math skills — keep practicing!"]