\(P(10) = -2(10)^2 + 40(10) - 50 = -200 + 400 - 50 = 150\)
["Understanding the Quadratic Equation: (P(10) = -2(10)^2 + 40(10) - 50) – A Step-by-Step Explanation", "Solving quadratic equations is a fundamental skill in algebra, essential for students, educators, and math enthusiasts. One such expression often used in practice and testing is:", "[\nP(10) = -2(10)^2 + 40(10) - 50\n]", "But what does this mean? Why does evaluating (P(10)) lead to the value 150? This article breaks down the expression step by step to explain how substitution and arithmetic yield this result, reinforcing key quadratic problem-solving techniques.", "---", "### What Is (P(10)) in This Equation?", "The function\n[\nP(x) = -2x^2 + 40x - 50\n]\nis a quadratic polynomial in standard form (ax^2 + bx + c). To find (P(10)), we substitute (x = 10) into the equation:", "[\nP(10) = -2(10)^2 + 40(10) - 50\n]", "---", "### Step-by-Step Evaluation", "Start by evaluating each term individually:", "#### 1. Compute ((10)^2):\n[\n10^2 = 100\n]", "Multiplying by (-2):\n[\n-2 \ imes 100 = -200\n]", "#### 2. Compute (40 \ imes 10):\n[\n40 \ imes 10 = 400\n]", "#### 3. The constant term is:\n[\n-50\n]", "---", "### Combine All Terms", "Now, substitute back into the equation:\n[\nP(10) = -200 + 400 - 50\n]", "Perform addition and subtraction step-by-step:\n[\n-200 + 400 = 200\n]\n[\n200 - 50 = 150\n]", "Thus,\n[\nP(10) = 150\n]", "---", "### Why Is This Problem Considered Important?", "Evaluating polynomials at specific values builds a concrete understanding of how changing input (x) affects output (P(x)). This form is commonly used in:", "- Solving real-world problems modeled by quadratics (e.g., projectile motion, profit maximization).\n- Assessing function behavior without graphing.\n- Preparing students for calculus and advanced algebra.", "---", "### Final Summary", "The calculation\n[\nP(10) = -2(10)^2 + 40(10) - 50 = -200 + 400 - 50 = 150\n]\ndemonstrates evaluating a quadratic function by substitution and arithmetic. Understanding this process strengthens algebraic fluency and supports application across scientific and technical fields.", "---", "### Key Takeaways:", "- Substitution: Always start by replacing (x) with the given value.\n- Order of Operations: Compute exponents first, then multiplication, followed by addition and subtraction.\n- Signs Matter: Double-check negative signs—here, (-2 \ imes 100 = -200) is critical.\n- Verification: Try plugging values in reverse or graphing to confirm correctness.", "---", "### Want to Practice?", "Try evaluating other quadratics at different (x)-values or explore how changing coefficients impacts (P(x)). Understanding these mechanisms deepens your grasp of polynomial functions.", "---", "Whether you’re learning algebra, teaching students, or brushing up your skills, mastering evaluations like (P(10) = 150) builds a strong foundation for more complex mathematical challenges.", "---", "Keywords: quadratic equation, evaluate (P(10)), polynomial evaluation, algebra 101, solving quadratics, (P(x) = -2x^2 + 40x - 50), step-by-step math, coordinate functions.", "Meta description: Learn how to evaluate (P(10) = -2(10)^2 + 40(10) - 50) step-by-step and understand the mechanics behind quadratic function evaluation."]
