Solution: Compute $ f(4) = 16 - 12 + m = 4 + m $. Compute $ g(4) = 16 - 12 + 2m = 4 + 2m $. Set $ 4 + m = 2(4 + 2m) $. Expand: $ 4 + m = 8 + 4m $. Rearrange: $ -4 = 3m $. Thus, $ m = - rac{4}{3} $.
["Solving for $ m $: A Step-by-Step Computation Explained", "Understanding how to solve simple algebraic equations is a fundamental skill in mathematics. This article walks through a clear and structured solution to the equation derived from two expressions evaluated at $ x = 4 $. The goal is to compute $ m $ by equating two linear expressions, simplified and solved step-by-step.", "---", "### The Problem", "We are given:", "- $ f(x) = x^2 - 12 + m $, evaluated at $ x = 4 $:\n $$\n f(4) = 16 - 12 + m = 4 + m\n $$", "- $ g(x) = x^2 - 12 + 2m $, evaluated at $ x = 4 $:\n $$\n g(4) = 16 - 12 + 2m = 4 + 2m\n $$", "We set $ f(4) = 2 \cdot g(4) $, leading to:", "$$\n4 + m = 2(4 + 2m)\n$$", "---", "### Step 1: Expand the Right-Hand Side", "Distribute the 2 on the right-hand side:", "$$\n4 + m = 8 + 4m\n$$", "---", "### Step 2: Rearrange the Equation", "Move all terms involving $ m $ to one side and constants to the other:", "$$\n4 + m - 4m = 8\n$$", "$$\n4 - 3m = 8\n$$", "---", "### Step 3: Solve for $ m $", "Subtract 4 from both sides:", "$$\n-3m = 4\n$$", "Divide both sides by $-3$:", "$$\nm = -\frac{4}{3}\n$$", "---", "### Final Answer", "We find that:", "$$\n\boxed{m = -\frac{4}{3}}\n$$", "---", "### Why This Matters in Math and Problem Solving", "This type of algebraic manipulation is not just about solving equations — it builds logical thinking, precision, and the ability to model real-life situations mathematically. Whether in physics, engineering, computer science, or everyday problem analysis, setting expressions equal and solving for unknowns is a powerful technique.", "Understanding each step clearly ensures that no detail is missed and reinforces foundational algebra skills essential for advanced mathematics.", "---", "### Key Takeaways", "- Substitute $ x = 4 $ into both expressions.\n- Set $ f(4) $ equal to twice $ g(4) $.\n- Expand and simplify step-by-step.\n- Isolate $ m $ using elementary algebra.\n- Verify your solution by plugging $ m $ back into the original expressions.", "By mastering such problems, you strengthen your mathematical fluency and confidence."]
