Question: If $ x + \frac{1}{x} = 4 $, what is the value of $ x^2 + \frac{1}{x^2} $?
["# Solving $ x + \frac{1}{x} = 4 $: How to Find $ x^2 + \frac{1}{x^2} $", "Mathematics often presents elegant problems with surprising connections. One classic question is: If $ x + \frac{1}{x} = 4 $, what is the value of $ x^2 + \frac{1}{x^2} $? This seemingly simple equation unlocks a powerful method used widely in algebra, calculus, and number theory. In this article, we’ll explore the step-by-step solution and explain why this technique matters.", "---", "## Understanding the Equation", "We begin with the given equation:", "$$\nx + \frac{1}{x} = 4\n$$", "Our goal is to compute:", "$$\nx^2 + \frac{1}{x^2}\n$$", "While $ x $ might seem abstract, the symmetry of the equation suggests we can derive $ x^2 + \frac{1}{x^2} $ without solving explicitly for $ x $. This approach avoids complicated algebra and leverages algebraic identities for efficiency and insight.", "---", "## Using Algebraic Identity", "A key algebraic identity connects expressions involving $ x + \frac{1}{x} $ to $ x^2 + \frac{1}{x^2} $:", "$$\n\left( x + \frac{1}{x} \right)^2 = x^2 + 2 + \frac{1}{x^2}\n$$", "This identity arises from expanding the square:", "$$\n\left( x + \frac{1}{x} \right)^2 = x^2 + 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^2} = x^2 + 2 + \frac{1}{x^2}\n$$", "Rearranging this gives:", "$$\nx^2 + \frac{1}{x^2} = \left( x + \frac{1}{x} \right)^2 - 2\n$$", "---", "## Substituting the Given Value", "We know from the problem:", "$$\nx + \frac{1}{x} = 4\n$$", "Square both sides:", "$$\n\left( x + \frac{1}{x} \right)^2 = 4^2 = 16\n$$", "Now substitute into the identity:", "$$\nx^2 + \frac{1}{x^2} = 16 - 2 = 14\n$$", "---", "## Final Answer", "$$\n\boxed{x^2 + \frac{1}{x^2} = 14}\n$$", "---", "## Why This Problem Matters", "This question is more than just a math drill — it demonstrates a foundational technique in algebra. Recognizing how to manipulate symmetric expressions enables quicker solutions across various fields:", "- Algebra & Equations: Deriving higher powers from low-degree terms.\n- Polynomials: Understanding roots and symmetric functions.\n- Calculus & Functions: Solving for expressions involving reciprocals.\n- Number Theory: Simplifying irrational expressions involving integers.", "Moreover, the method is generalizable: if $ y = x + \frac{1}{x} $, then $ y^2 - 2 = x^2 + \frac{1}{x^2} $. This shortcut is invaluable in advanced problem-solving.", "---", "## Step-by-Step Summary", "1. Given:\n $$\n x + \frac{1}{x} = 4\n $$\n2. Square both sides:\n $$\n \left( x + \frac{1}{x} \right)^2 = 16\n $$\n3. Apply identity:\n $$\n x^2 + \frac{1}{x^2} = 16 - 2 = 14\n $$\n4. Final result:\n $$\n \boxed{x^2 + \frac{1}{x^2} = 14}\n $$", "---", "## Further Exploration", "Want to master similar techniques? Practice transforming expressions like $ x^n + \frac{1}{x^n} $ using recurrence relations based on powers of $ x + \frac{1}{x} $. With consistent practice, you’ll uncover elegant solutions hidden in complex equations.", "This question — If $ x + \frac{1}{x} = 4 $, what is $ x^2 + \frac{1}{x^2} $? — perfectly illustrates how simple forms unlock powerful mathematical tools."]
