5Question: Find the quadratic polynomial $ h(x) $ such that $ h(1) = 5 $, $ h(-1) = 3 $, and the sum of its roots is $ 4 $.
["Title: How to Find the Quadratic Polynomial $ h(x) $ Given Its Root Sum and Two Function Values", "Meta Description:\nLearn how to determine the quadratic polynomial $ h(x) $ when given $ h(1) = 5 $, $ h(-1) = 3 $, and the sum of its roots equals $ 4 $. We walk through the steps using polynomial properties and system solving.", "---", "### Finding the Quadratic Polynomial $ h(x) $: Key Information", "Suppose the quadratic polynomial has the general form:\n[\nh(x) = ax^2 + bx + c\n]\nWe are given three conditions:\n1. $ h(1) = 5 $\n2. $ h(-1) = 3 $\n3. The sum of the roots is $ 4 $", "Our goal is to find the coefficients $ a $, $ b $, and $ c $.", "---", "### Step 1: Use the given values to form equations", "Using $ h(1) = 5 $:\n[\na(1)^2 + b(1) + c = 5 \quad \Rightarrow \quad a + b + c = 5 \quad \ ext{(Equation 1)}\n]", "Using $ h(-1) = 3 $:\n[\na(-1)^2 + b(-1) + c = 3 \quad \Rightarrow \quad a - b + c = 3 \quad \ ext{(Equation 2)}\n]", "---", "### Step 2: Use sum of roots to relate $ b $ and $ a $", "For any quadratic equation $ h(x) = ax^2 + bx + c $, the sum of the roots is:\n[\n-\frac{b}{a} = 4 \quad \Rightarrow \quad b = -4a \quad \ ext{(Equation 3)}\n]", "---", "### Step 3: Solve the system of equations", "Substitute Equation 3 ($ b = -4a $) into Equations 1 and 2.", "Substituting into Equation 1:\n[\na + (-4a) + c = 5 \quad \Rightarrow \quad -3a + c = 5 \quad \ ext{(Equation 4)}\n]", "Substituting into Equation 2:\n[\na - (-4a) + c = 3 \quad \Rightarrow \quad a + 4a + c = 3 \quad \Rightarrow \quad 5a + c = 3 \quad \ ext{(Equation 5)}\n]", "Now solve Equations 4 and 5:\nFrom Equation 4:\n[\nc = 5 + 3a\n]\nSubstitute into Equation 5:\n[\n5a + (5 + 3a) = 3 \quad \Rightarrow \quad 8a + 5 = 3 \quad \Rightarrow \quad 8a = -2 \quad \Rightarrow \quad a = -\frac{1}{4}\n]", "Now find $ b $:\n[\nb = -4a = -4 \left(-\frac{1}{4}\right) = 1\n]", "And $ c $:\n[\nc = 5 + 3\left(-\frac{1}{4}\right) = 5 - \frac{3}{4} = \frac{17}{4}\n]", "---", "### Step 4: Write the polynomial", "[\nh(x) = -\frac{1}{4}x^2 + x + \frac{17}{4}\n]", "---", "### Step 5: Verify the conditions", "- $ h(1) = -\frac{1}{4}(1)^2 + 1 + \frac{17}{4} = -\frac{1}{4} + \frac{4}{4} + \frac{17}{4} = \frac{20}{4} = 5 $ ✓\n- $ h(-1) = -\frac{1}{4}(1) - 1 + \frac{17}{4} = -\frac{1}{4} - \frac{4}{4} + \frac{17}{4} = \frac{12}{4} = 3 $ ✓\n- Sum of roots: $ -\frac{b}{a} = -\frac{1}{-\frac{1}{4}} = 4 $ ✓", "---", "Conclusion:\nThe quadratic polynomial satisfying the given conditions is\n[\n\boxed{h(x) = -\frac{1}{4}x^2 + x + \frac{17}{4}}\n]\nThis example demonstrates how combining functional values and root properties allows precise determination of unknown coefficients—critical in polynomial fitting and applied mathematics.", "Keywords: quadratic polynomial, find $ h(x) $, h(1)=5, h(-1)=3, sum of roots 4, polynomial roots, solving quadratic polynomials, system of equations", "---", "For further learning, practice with different sum constraints and functional values—mastering these tools enables elegant solutions across algebra, physics, and engineering problems.", "---", "Related Articles:\n- How to Determine Root Sum and Product from Coefficients\n- Solving Quadratic Polynomials: From Roots to General Form\n- Using Function Values to Find Missing Coefficients in Quadratics", "Tagline: Master quadratics—understand structure, apply logic, verify results instantly."]
