At \( t = 4 \), \( v(4) = 3(4)^2 - 12(4) + 9 \).
["# At ( t = 4 ), ( v(4) = 3(4)^2 - 12(4) + 9 ): A Step-by-Step Evaluation", "Mathematics often involves evaluating expressions at specific values, especially in physics and engineering contexts. One common example is computing velocity or position functions at a given time ( t ). In this article, we explore the expression\n[ v(4) = 3(4)^2 - 12(4) + 9 ]\nand break down the calculation step-by-step to find the exact value.", "---", "## Understanding the Expression", "The expression\n[ v(4) = 3(4)^2 - 12(4) + 9 ]\nmodels a velocity function evaluated at time ( t = 4 ). This type of quadratic expression frequently appears in kinematics when analyzing motion with constant acceleration.", "We rewrite it clearly:\n[ v(4) = 3 \cdot (4)^2 - 12 \cdot 4 + 9 ]", "---", "## Step 1: Evaluate the exponent\nWe begin with the squared term:\n[ (4)^2 = 16 ]\nNow multiply by 3:\n[ 3 \cdot 16 = 48 ]", "---", "## Step 2: Evaluate the linear term\nNext, calculate:\n[ 12 \cdot 4 = 48 ]", "---", "## Step 3: Combine all terms\nSubstituting back, we have:\n[ v(4) = 48 - 48 + 9 ]", "Simplify:\n[ 48 - 48 = 0 ]\n[ 0 + 9 = 9 ]", "---", "## Final Result", "Therefore,\n[ v(4) = 9 ]", "This means at time ( t = 4 ), the value of the velocity function is exactly 9 units per time interval.", "---", "## Why This Matters in Real-World Applications", "Evaluating such expressions at precise values is essential in physics for determining instantaneous velocity, position, or other dynamic behaviors. Simple polynomial calculations like this form the foundation for analyzing motion, optimizing systems, and solving real-life engineering problems.", "---", "## Conclusion", "Evaluating ( v(4) = 3(4)^2 - 12(4) + 9 ) step-by-step confirms that:\n[ \mathbf{v(4) = 9} ]\nThis straightforward computation reinforces the importance of careful arithmetic and understanding function evaluation—key skills for students, physicists, and engineers alike.", "---", "## Key Takeaways", "- Break down each term carefully: squaring, multiplication, and addition/subtraction.\n- Use parentheses to ensure correct order of operations.\n- This expression exemplifies quadratic velocity functions common in kinematic equations.\n- Always verify calculations step-by-step to avoid errors.", "---", "Tagline: Mastering algebra at key points like ( t = 4 ) transforms abstract formulas into practical insights—start evaluating today!"]
