\( x = \frac{-460 \pm 509.5}{8} \)

["Understanding the Equation: ( x = \dfrac{-460 \pm 509.5}{8} )", "When solving equations involving a ( \pm ) symbol, the expression typically represents two distinct solutions derived from a single fractional equation. The equation:", "[\nx = \frac{-460 \pm 509.5}{8}\n]", "can be understood as two separate values that solve for ( x ), reflecting variation in sign before division.", "---", "### Breaking Down the Equation", "The expression uses the "plus-minus" convention (( \pm )) common in algebra, meaning two results:", "- Positive case: Use ( +509.5 )\n- Negative case: Use ( -509.5 )", "Both are divided by 8.", "---", "### Computing Each Solution", "1. First solution (using ( +509.5 )):", "[\nx_1 = \frac{-460 + 509.5}{8} = \frac{49.5}{8} = 6.1875\n]", "2. Second solution (using ( -509.5 )):", "[\nx_2 = \frac{-460 - 509.5}{8} = \frac{-969.5}{8} = -121.1875\n]", "---", "### Practical Applications and Why It Matters", "Equations with a ( \pm ) form are frequently used in physics, engineering, and economics to model outcomes with uncertainty or symmetric variation around a central value. For example:", "- Physics: When calculating displacement with symmetric motion or wave motion amplitudes.\n- Engineering: To analyze tolerances and variations in mechanical components.\n- Finance: Modeling profit or loss scenarios with a range of possible outcomes.", "Understanding how to properly handle such expressions ensures accurate computation and correct interpretation of results.", "---", "### Step-by-Step Summary", "- Start with: ( x = \frac{-460 \pm 509.5}{8} )\n- Separate into two solutions:\n 1. ( x = \frac{-460 + 509.5}{8} = 6.1875 )\n 2. ( x = \frac{-460 - 509.5}{8} = -121.1875 )\n- Interpret as symmetric deviations from the base value ( \frac{-460}{8} = -57.5 ):\n - ( x_1 = -57.5 + 63.9375 = 6.1875 )\n - ( x_2 = -57.5 - 63.9375 = -121.1875 )", "---", "### Final Thoughts", "Learning how to expand equations involving ( x = \frac{-a \pm b}{c} ) equips you with a powerful algebraic tool for solving real-world problems. Mastering both the positive and negative cases ensures precision and deeper comprehension in advanced problem-solving.", "---", "Keywords: ( x = \frac{-460 \pm 509.5}{8} ), solving linear equations, algebra tips, mathematical methodology, solving quadratic context equations, step-by-step algebra, real-world equation applications.", "---", "Need more help with equations? Explore our full range of guides on algebraic solutions, equations with ( \pm ), and practical math applications!"]