Y = -2(6)^2 + 24(6) + 100 = -2(36) + 144 + 100 = -72 + 144 + 100 = 172

["Final Calculation and Explanation of the Quadratic Equation Y = -2(6)² + 24(6) + 100", "When solving quadratic equations, accurate arithmetic and clear algebraic manipulation are essential—especially in contexts like physics, economics, or engineering where such formulas model real-world phenomena. One such expression commonly encountered is:", "[ Y = -2(6)^2 + 24(6) + 100 ]", "Let’s break this down step-by-step to verify the simplified result and understand the significance of the expression.", "---", "### Step-by-Step Evaluation", "Start with the original equation:\n[\nY = -2(6)^2 + 24(6) + 100\n]", "Step 1: Evaluate the exponent\n[\n(6)^2 = 36\n]\nSubstitute into the equation:\n[\nY = -2(36) + 24(6) + 100\n]", "Step 2: Perform multiplications\n[\n-2 \ imes 36 = -72\n]\n[\n24 \ imes 6 = 144\n]\nNow the equation becomes:\n[\nY = -72 + 144 + 100\n]", "Step 3: Add the terms sequentially\n[\n-72 + 144 = 72\n]\n[\n72 + 100 = 172\n]", "Thus,\n[\nY = 172\n]", "---", "### Understanding the Quadratic Formula", "The expression\n[\nY = -2(6)^2 + 24(6) + 100\n]\nis a quadratic equation in standard form ( Y = ax^2 + bx + c ), where:\n- ( a = -2 )\n- ( b = 24 )\n- ( c = 100 )\nand ( x = 6 )", "This specific evaluation reflects a real-world application, such as calculating profit at a sales threshold, modeling projectile motion under gravity (adjusted for acceleration), or analyzing parabolic cost functions.", "Using the quadratic formula:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]\nSubstituting values gives:\n[\nx = \frac{-24 \pm \sqrt{24^2 - 4(-2)(100)}}{2(-2)} = \frac{-24 \pm \sqrt{576 + 800}}{-4} = \frac{-24 \pm \sqrt{1376}}{-4}\n]\nWhile simplifying this further yields the root(s) of the equation, here we focused on evaluating the expression at ( x = 6 ).", "---", "### Significance and Context", "Knowing the value of such quadratic expressions is crucial in fields like finance (where revenue models follow quadratic trends), physics (e.g., motion under constant acceleration), and operations research (optimal point analysis). Being able to compute and interpret these values efficiently ensures accurate forecasting, design, and decision-making.", "---", "### Conclusion", "The expression ( Y = -2(6)^2 + 24(6) + 100 ) simplifies to ( \mathbf{172} ), a clear demonstration of combining exponents, multiplication, and addition in a real-valued quadratic context. Whether teaching algebra fundamentals or solving applied problems, mastering this step-by-step process builds confidence in manipulating mathematical models with precision.", "If you frequently encounter such equations, consider practicing with different coefficients to strengthen algebraic fluency and expand your ability to interpret and apply quadratic relationships.", "---", "Keywords: quadratic equation calculation, algebraic simplification, solving Y = ax² + bx + c, evaluate quadratic expression, mathematical evaluation, real-world equations, exponential and linear operations in algebra.", "---", "Understanding core calculations like ( Y = -2(6)^2 + 24(6) + 100 ) empowers learners and professionals to decode complex models with clarity and accuracy."]