Question: If $x + rac{1}{x} = 4$, what is the value of $x^2 + rac{1}{x^2}$?
["SEO-Optimized Article", "Question: If $ x + \frac{1}{x} = 4 $, What Is the Value of $ x^2 + \frac{1}{x^2} $?", "Mathematics often presents elegant challenges, and one such intriguing problem is finding the value of $ x^2 + \frac{1}{x^2} $ given that $ x + \frac{1}{x} = 4 $. This question not only tests algebraic manipulation skills but also reveals a powerful identity that simplifies its solution.", "### Understanding the Problem", "We are given:\n[\nx + \frac{1}{x} = 4\n]\nThe goal is to compute:\n[\nx^2 + \frac{1}{x^2}\n]", "Directly squaring both sides of the given equation helps unlock the hidden expression efficiently.", "### Step-by-Step Solution", "Start by squaring both sides of the equation:\n[\n\left( x + \frac{1}{x} \right)^2 = 4^2\n]\n[\nx^2 + 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^2} = 16\n]", "Simplify the middle term:\n[\nx^2 + 2 + \frac{1}{x^2} = 16\n]", "Now, subtract 2 from both sides:\n[\nx^2 + \frac{1}{x^2} = 16 - 2 = 14\n]", "### Final Answer", "Thus, if $ x + \frac{1}{x} = 4 $, then:\n[\nx^2 + \frac{1}{x^2} = 14\n]", "### Why This Works", "The key identity used is:\n[\n\left( x + \frac{1}{x} \right)^2 = x^2 + 2 + \frac{1}{x^2}\n]\nThis relationship always holds for nonzero $ x $, allowing transformation from a linear combination to a quadratic expression.", "This method saves time and avoids complicated substitutions, making it a preferred technique in algebra problems.", "### Related Keywords for SEO Optimization\n- $ x + \frac{1}{x} = 4 $ solution\n- $ x^2 + \frac{1}{x^2} $ formula\n- algebraic identities\n- working with reciprocal equations\n- fast way to find $ x^2 + \frac{1}{x^2} $\n- solve quadratic expressions from symmetric sums", "---", "By applying a simple squaring technique and well-known identities, we efficiently solve the problem and reinforce fundamental algebraic skills useful for students and math enthusiasts alike."]
