Question: The functions $ f(x) = x^2 - 6x + m $ and $ g(x) = x^2 - 6x + 3m $ are evaluated at $ x = 5 $. If $ f(5) = 2g(5) $, find $ m $.
["SEO-Optimized Article: Solving $ f(5) = 2g(5) $ for $ m $ Using Quadratic Functions", "Understanding how to solve equations involving quadratic functions is essential for mastering algebra. In this article, we explore a common problem involving two related quadratic functions: $ f(x) = x^2 - 6x + m $ and $ g(x) = x^2 - 6x + 3m $. The condition $ f(5) = 2g(5) $ creates a real-world context for learning function evaluation and equation solving. We’ll walk through the step-by-step process and explain key algebraic concepts along the way.", "---", "### Background: Evaluating Functions at a Point", "Functions like $ f(x) $ and $ g(x) $ assign output values based on input $ x $. Here, both functions share the same structure but differ by a constant multiple in the constant term. Evaluating them at $ x = 5 $ allows us to form an equation that relates their outputs — a powerful technique for finding unknown parameters like $ m $.", "---", "### Step 1: Substitute $ x = 5 $ into both functions", "Start by calculating $ f(5) $:", "$$\nf(5) = (5)^2 - 6(5) + m = 25 - 30 + m = -5 + m\n$$", "Next, compute $ g(5) $:", "$$\ng(5) = (5)^2 - 6(5) + 3m = 25 - 30 + 3m = -5 + 3m\n$$", "---", "### Step 2: Apply the given condition $ f(5) = 2g(5) $", "Substitute the expressions:", "$$\n-5 + m = 2(-5 + 3m)\n$$", "Now simplify the right-hand side:", "$$\n-5 + m = -10 + 6m\n$$", "---", "### Step 3: Solve for $ m $", "Bring all terms involving $ m $ to one side and constants to the other:", "$$\n-5 + m + 10 = 6m\n\Rightarrow 5 + m = 6m\n\Rightarrow 5 = 5m\n\Rightarrow m = 1\n$$", "---", "### Why This Matters", "This problem illustrates how constants in quadratic functions influence outputs when evaluated at a specific point — a concept useful in physics, engineering, and optimization tasks. Knowing how to manipulate equations involving such functions helps students and professionals alike solve real-life modeling problems.", "---", "### Summary", "To solve $ f(5) = 2g(5) $ for $ m $:", "1. Evaluate $ f(5) = -5 + m $\n2. Evaluate $ g(5) = -5 + 3m $\n3. Substitute into $ f(5) = 2g(5) $\n4. Solve the resulting linear equation: $ -5 + m = 2(-5 + 3m) $\n5. Simplify and isolate $ m $ to find $ m = 1 $", "---", "Keywords: quadratic functions, evaluate functions, solve for m, f(x) = x² – 6x + m, g(x) = x² – 6x + 3m, algebra problem, function equations, solve linear equation, math problem solving.", "Meta Description: Learn how to solve for $ m $ in quadratic functions using function evaluation at a point. Example: Given $ f(x) = x^2 - 6x + m $, $ g(x) = x^2 - 6x + 3m $, and $ f(5) = 2g(5) $, find $ m $ step by step.", "---", "Continue practicing with similar problems to strengthen your algebraic skills!"]
