Solution: Let $ h(x) = ax^2 + bx + c $. From $ h(1) = a + b + c = 5 $ and $ h(-1) = a - b + c = 3 $, adding gives $ 2a + 2c = 8 $, so $ a + c = 4 $. The sum of roots is $ - rac{b}{a} = 4 $, so $ b = -4a $. Substituting $ b = -4a $ into $ a + b + c = 5 $: $ a - 4a + c = 5 $ â $ -3a + c = 5 $. Since $ a + c = 4 $, subtracting gives $ -4a = 1 $, so $ a = - rac{1}{4} $. Then $ c = 4 - a = 4 + rac{1}{4} = rac{17}{4} $, and $ b = -4a = 1 $. Thus, $ h(x) = - rac{1}{4}x^2 + x + rac{17}{4} $. Multip
["Solving Quadratic Functions: Deriving the Exact Form of $ h(x) = ax^2 + bx + c $ Using Given Conditions", "Understanding how to derive a quadratic function from key properties—like function values and root behavior—is a foundational skill in algebra. Consider the general quadratic function:\n$$\nh(x) = ax^2 + bx + c\n$$\nWe are given:\n- $ h(1) = a(1)^2 + b(1) + c = a + b + c = 5 $\n- $ h(-1) = a(-1)^2 + b(-1) + c = a - b + c = 3 $", "Adding these two equations:\n$$\n(a + b + c) + (a - b + c) = 5 + 3 \Rightarrow 2a + 2c = 8 \Rightarrow a + c = 4\n$$", "Next, the sum of the roots of a quadratic $ ax^2 + bx + c = 0 $ is $ -\frac{b}{a} $. Given this sum is $ 4 $, we have:\n$$\n-\frac{b}{a} = 4 \Rightarrow b = -4a\n$$", "Now substitute $ b = -4a $ into $ a + b + c = 5 $:\n$$\na - 4a + c = 5 \Rightarrow -3a + c = 5\n$$\nBut we already know $ a + c = 4 $. Subtract the equations:\n$$\n(-3a + c) - (a + c) = 5 - 4 \Rightarrow -4a = 1 \Rightarrow a = -\frac{1}{4}\n$$", "From $ a + c = 4 $,\n$$\nc = 4 - \left(-\frac{1}{4}\right) = 4 + \frac{1}{4} = \frac{17}{4}\n$$\nAnd $ b = -4a = -4 \left(-\frac{1}{4}\right) = 1 $", "Thus, the quadratic is:\n$$\nh(x) = -\frac{1}{4}x^2 + x + \frac{17}{4}\n$$", "To eliminate fractions, multiply the entire expression by 4:\n$$\nh(x) = -x^2 + 4x + 17\n$$", "Each condition checks out:\n- $ h(1) = -1 + 4 + 17 = 20 $? Wait—correction:\n$$\nh(1) = -\frac{1}{4}(1)^2 + 1 + \frac{17}{4} = -\frac{1}{4} + 1 + \frac{17}{4} = \left(-1 + 4 + 17\right)/4 = 20/4 = 5 \quad \ ext{(correct)}\n$$\n$$\nh(-1) = -\frac{1}{4}(1) - 1 + \frac{17}{4} = -\frac{1}{4} - 1 + \frac{17}{4} = \frac{-1 -4 + 17}{4} = \frac{12}{4} = 3 \quad \ ext{(correct)}\n$$\nSum of roots: $ -\frac{b}{a} = -\frac{1}{-1/4} = 4 $ — matches.", "Therefore, the precise expression for $ h(x) $ satisfying all conditions is:\n$$\n\boxed{h(x) = -x^2 + 4x + \dfrac{17}{4}} \quad \ ext{or} \quad \boxed{-\dfrac{1}{4}x^2 + x + \dfrac{17}{4}}\n$$", "This systematic approach—using function values, sum of roots, and coefficient relationships—forms a powerful method for solving and deriving quadratic equations efficiently.", "---", "SEO Keywords: quadratic function derivation, solve quadratic with conditions, use function values to find coefficients, algebra quadratic solution, derive $ h(x) = ax^2 + bx + c $, sum of roots from coefficients, eliminate fractions in quadratics, algebra problem-solving.", "---", "Meta Description:\nDiscover how to derive the exact form of a quadratic function using given values and root properties. Step-by-step solution for $ h(x) = ax^2 + bx + c $ with $ h(1)=5 $, $ h(-1)=3 $, and sum of roots $ 4 $, including verification and simplified version."]
