Question: The functions $ f(x) = x^2 - 6x + m $ and $ g(x) = x^2 - 6x + 5m $ are evaluated at $ x = 3 $. If $ f(3) = 2g(3) $, find $ m $.
["Understanding the Relationship Between Two Quadratic Functions at $ x = 3 $", "When studying quadratic functions, evaluating them at key points often reveals important relationships—especially when comparing outputs. In this article, we explore a specific scenario involving two quadratic functions evaluated at $ x = 3 $, linked by a condition: $ f(3) = 2g(3) $. By solving for the parameter $ m $, we uncover insights into function behavior and algebraic problem-solving.", "---", "### Setting Up the Functions", "We are given two quadratic functions:\n$$\nf(x) = x^2 - 6x + m\n$$\n$$\ng(x) = x^2 - 6x + 5m\n$$\nBoth functions share the same quadratic and linear terms but differ in their constant terms: $ m $ vs. $ 5m $. We evaluate these at $ x = 3 $.", "---", "### Evaluating $ f(3) $", "Substitute $ x = 3 $ into $ f(x) $:\n$$\nf(3) = (3)^2 - 6(3) + m = 9 - 18 + m = -9 + m\n$$", "---", "### Evaluating $ g(3) $", "Substitute $ x = 3 $ into $ g(x) $:\n$$\ng(3) = (3)^2 - 6(3) + 5m = 9 - 18 + 5m = -9 + 5m\n$$", "---", "### Applying the Given Condition", "The problem states that $ f(3) = 2g(3) $. Substituting the expressions:\n$$\n-9 + m = 2(-9 + 5m)\n$$\nSimplify the right-hand side:\n$$\n-9 + m = -18 + 10m\n$$", "Now solve for $ m $:\nBring all terms involving $ m $ to one side and constants to the other:\n$$\n-9 + 18 = 10m - m\n$$\n$$\n9 = 9m\n$$\n$$\nm = 1\n$$", "---", "### Verifying the Solution", "To ensure correctness, substitute $ m = 1 $ back into both functions:\n- $ f(3) = -9 + 1 = -8 $\n- $ g(3) = -9 + 5(1) = -4 $\nCheck the condition: $ f(3) = 2g(3) \Rightarrow -8 = 2(-4) $, which holds true.", "---", "### Why This Matters in Algebra and Problem Solving", "This problem illustrates how to work with function values at specific points and set up proportional relationships. It emphasizes:", "- Accurate substitution into polynomial expressions\n- Manipulation of linear and constant terms\n- Solving equations with one unknown efficiently", "Such skills are foundational in algebra, supporting deeper understanding of quadratic behavior, graphing, and real-world modeling.", "---", "### Final Thoughts", "Solving for $ m $ in function comparisons at a particular $ x $-value sharpens critical thinking and algebraic precision. Whether in exams, coding, or applied mathematics, mastering these steps builds confidence in tackling abstract mathematical relationships.", "---", "Conclusion\nFor the functions $ f(x) = x^2 - 6x + m $ and $ g(x) = x^2 - 6x + 5m $, the condition $ f(3) = 2g(3) $ leads directly to $ m = 1 $. This elegant solution demonstrates how targeted evaluation and algebraic manipulation uncover the value of an unknown parameter."]
