Question: Find the quadratic polynomial $ p(x) $ such that $ p(1) = 4 $, $ p(2) = 11 $, and $ p(3) = 22 $.
["# Find the Quadratic Polynomial $ p(x) $ Given $ p(1) = 4 $, $ p(2) = 11 $, and $ p(3) = 22 $", "When asked to find a quadratic polynomial $ p(x) $ based on specific output values at three distinct points, we are essentially solving a pattern recognition problem with constraints. This article walks you through the process of determining $ p(x) = ax^2 + bx + c $ using the given conditions:", "### Given Conditions\nWe are given:\n- $ p(1) = 4 $\n- $ p(2) = 11 $\n- $ p(3) = 22 $", "Since $ p(x) $ is quadratic, it can be expressed in the standard form:\n$$\np(x) = ax^2 + bx + c\n$$", "### Step 1: Set Up the System of Equations\nSubstituting the known values into the polynomial form yields a system of three equations:", "1. $ p(1) = a(1)^2 + b(1) + c = 4 \Rightarrow a + b + c = 4 $\n2. $ p(2) = a(2)^2 + b(2) + c = 11 \Rightarrow 4a + 2b + c = 11 $\n3. $ p(3) = a(3)^2 + b(3) + c = 22 \Rightarrow 9a + 3b + c = 22 $", "### Step 2: Solve the System of Equations", "We now solve:\n$$\n\begin{cases}\na + b + c = 4 \quad \ ext{(1)} \\n4a + 2b + c = 11 \quad \ ext{(2)} \\n9a + 3b + c = 22 \quad \ ext{(3)}\n\end{cases}\n$$", "Subtract equation (1) from equation (2):\n$$\n(4a + 2b + c) - (a + b + c) = 11 - 4 \Rightarrow 3a + b = 7 \quad \ ext{(4)}\n$$", "Subtract equation (2) from equation (3):\n$$\n(9a + 3b + c) - (4a + 2b + c) = 22 - 11 \Rightarrow 5a + b = 11 \quad \ ext{(5)}\n$$", "Now subtract equation (4) from equation (5):\n$$\n(5a + b) - (3a + b) = 11 - 7 \Rightarrow 2a = 4 \Rightarrow a = 2\n$$", "Substitute $ a = 2 $ into equation (4):\n$$\n3(2) + b = 7 \Rightarrow 6 + b = 7 \Rightarrow b = 1\n$$", "Substitute $ a = 2 $, $ b = 1 $ into equation (1):\n$$\n2 + 1 + c = 4 \Rightarrow c = 1\n$$", "### Step 3: Final Polynomial\nSubstitute $ a = 2 $, $ b = 1 $, $ c = 1 $ into the standard form:\n$$\np(x) = 2x^2 + x + 1\n$$", "### Step 4: Verification\nCheck against given values:\n- $ p(1) = 2(1)^2 + 1 + 1 = 4 $ ✅\n- $ p(2) = 2(4) + 2 + 1 = 8 + 2 + 1 = 11 $ ✅\n- $ p(3) = 2(9) + 3 + 1 = 18 + 3 + 1 = 22 $ ✅", "All conditions are satisfied.", "### Why This Method Works\nThis approach leverages the fact that a quadratic polynomial is uniquely determined by three distinct points. By forming a system of linear equations based on known values and solving step-by-step, we ensure accuracy and clarity.", "---", "### Advanced Insight: Lagrange Interpolation\nAlternatively, we can use Lagrange interpolation to construct $ p(x) $ directly:\n$$\np(x) = 4 \cdot \frac{(x-2)(x-3)}{(1-2)(1-3)} + 11 \cdot \frac{(x-1)(x-3)}{(2-1)(2-3)} + 22 \cdot \frac{(x-1)(x-2)}{(3-1)(3-2)}\n$$", "Calculating each term confirms the same result:\n$$\np(x) = 2x^2 + x + 1\n$$", "---", "### Conclusion\nFinding the quadratic polynomial $ p(x) $ satisfying $ p(1) = 4 $, $ p(2) = 11 $, and $ p(3) = 22 $ is a straightforward application of system solving. The final polynomial is:", "$$\n\boxed{p(x) = 2x^2 + x + 1}\n$$", "This method applies to any similar interpolation problem, making it a cornerstone technique in algebra and applied mathematics.", "---", "Keywords: quadratic polynomial, interpolation, solve for $ p(x) $, $ p(1) = 4 $, $ p(2) = 11 $, $ p(3) = 22 $, $ ax^2 + bx + c $, system of equations, Lagrange interpolation", "Meta Description: Find the quadratic polynomial $ p(x) = ax^2 + bx + c $ such that $ p(1) = 4 $, $ p(2) = 11 $, and $ p(3) = 22 $. Step-by-step solution with verification."]
