\( P = -2(3)^2 + 12(3) - 5 = -18 + 36 - 5 = 13 \).
["# Solving –2(3)² + 12(3) – 5 Step-by-Step: Proving P = 13", "Mastering quadratic equations is essential for students and math enthusiasts alike. One key expression combining exponentiation and linear terms is ( P = -2(3)^2 + 12(3) - 5 ). In this article, we’ll break down the calculation ( P = -2(3)^2 + 12(3) - 5 ) thoroughly, simplify it step-by-step, and confirm that ( P = 13 ). We’ll also explain how to solve quadratic expressions like this efficiently, making it easier to solve real-world problems involving parabolas, physics, and optimization.", "## Understanding the Expression", "The expression ( P = -2(3)^2 + 12(3) - 5 ) involves order of operations governed by PEMDAS/BODMAS rules: exponents first, then multiplication, followed by addition and subtraction.", "## Step-by-Step Calculation", "Let’s evaluate each term carefully:", "### Step 1: Evaluate the exponent\n( (3)^2 = 9 )\nNow substitute into the expression:\n[ P = -2(9) + 12(3) - 5 ]", "### Step 2: Perform multiplication\nCalculate each term individually:\n- ( -2 \ imes 9 = -18 )\n- ( 12 \ imes 3 = 36 )", "Now the expression becomes:\n[ P = -18 + 36 - 5 ]", "### Step 3: Carry out addition and subtraction left to right\n- First, ( -18 + 36 = 18 )\n- Then, ( 18 - 5 = 13 )", "### Final Result\n[ P = \boxed{13} ]", "---", "## Why Solving Quadratics Like This Matters", "While this specific quadratic simplifies to a constant, understanding how to compute expressions like ( -2b^2 + cb - d ) is foundational. Such forms commonly appear in:", "- Parabolic modeling in physics (projectile motion, optimization)\n- Economics (profit maximization functions)\n- Geometry (calculating distances and areas)", "Breaking down each operation step-by-step helps prevent calculation errors and builds fluency in manipulating algebraic expressions—crucial for more complex equations.", "---", "## How to Solve Quadratic Equations Efficiently", "1. Rewrite standard form: ( ax^2 + bx + c = 0 )\n2. Substitute values correctly following PEMDAS\n3. Simplify operations (exponents first, then multiplications, then addition/subtraction)\n4. Check your work by plugging the solution back into the original equation", "This approach ensures accuracy and reduces confusion, especially when handling negative coefficients or large numbers like in ( -2(3)^2 ).", "---", "## Practice Problem: Verify Your Skills", "Try computing:\n[ Q = -3(4)^2 + 14(4) - 7 ]\nCan you identify each step and confirm ( Q = ? )? This reinforces your ability to evaluate quadratic expressions efficiently.", "---", "## Conclusion", "The evaluation of ( P = -2(3)^2 + 12(3) - 5 ) shows that while some quadratic expressions simplify neatly to constants, mastering order of operations and careful substitution is key. Whether solving problems in math, science, or engineering, precise step-by-step computation empowers confident problem-solving.", "Remember:\n( P = -2(9) + 36 - 5 = -18 + 36 - 5 = 13 ).", "Keep practicing—mastery comes with consistent effort!"]
