Solution: Assume $ V(t) = at^2 + bt + c $. From $ V(1) = a + b + c = 120 $, $ V(2) = 4a + 2b + c = 200 $, $ V(3) = 9a + 3b + c = 300 $. Subtract first equation from the second: $ 3a + b = 80 $. Subtract second from the third: $ 5a + b = 100 $. Subtract these: $ 2a = 20 $ → $ a = 10 $. Then $ 3(10) + b = 80 $ → $ b = 50 $. From $ a + b + c = 120 $: $ 10 + 50 + c = 120 $ → $ c = 60 $. Thus, $ V(t) = 10t^2 + 50t + 60 $. For $ t = 4 $: $ V(4) = 10(16) + 50(4) + 60 = 160 + 200 + 60 = 420 $. Fin

["Solution to Quadratic Function Using System of Equations: Find V(4) Given Key Values", "Understanding and solving quadratic functions is essential in many fields, from physics to economics. In this article, we walk through a precise mathematical solution using a quadratic model ( V(t) = at^2 + bt + c ), based on three known data points, to accurately determine the output at ( t = 4 ).", "---", "### Step 1: Set up the system of equations", "We are given:", "- ( V(1) = a(1)^2 + b(1) + c = a + b + c = 120 )  (Equation 1)\n- ( V(2) = a(2)^2 + b(2) + c = 4a + 2b + c = 200 ) (Equation 2)\n- ( V(3) = a(3)^2 + b(3) + c = 9a + 3b + c = 300 ) (Equation 3)", "These represent a system of linear equations derived directly from the quadratic model.", "---", "### Step 2: Eliminate ( c ) by subtracting equations", "Subtract Equation 1 from Equation 2:", "[\n(4a + 2b + c) - (a + b + c) = 200 - 120 \implies 3a + b = 80 \quad \ ext{(Equation 4)}\n]", "Then subtract Equation 2 from Equation 3:", "[\n(9a + 3b + c) - (4a + 2b + c) = 300 - 200 \implies 5a + b = 100 \quad \ ext{(Equation 5)}\n]", "---", "### Step 3: Solve for ( a )", "Subtract Equation 4 from Equation 5:", "[\n(5a + b) - (3a + b) = 100 - 80 \implies 2a = 20 \implies a = 10\n]", "---", "### Step 4: Solve for ( b )", "Substitute ( a = 10 ) into Equation 4:", "[\n3(10) + b = 80 \implies 30 + b = 80 \implies b = 50\n]", "---", "### Step 5: Solve for ( c )", "Substitute ( a = 10 ) and ( b = 50 ) into Equation 1:", "[\n10 + 50 + c = 120 \implies c = 60\n]", "---", "### Final quadratic function", "[\nV(t) = 10t^2 + 50t + 60\n]", "---", "### Step 6: Evaluate at ( t = 4 )", "[\nV(4) = 10(4)^2 + 50(4) + 60 = 10(16) + 200 + 60 = 160 + 200 + 60 = 420\n]", "---", "### Final Answer", "[\n\boxed{420}\n]", "This structured solution demonstrates how subtracting equations eliminates variables and simplifies finding coefficients, leading to accurate prediction of the quadratic function’s behavior at any point in time."]