Question: If \( x + rac{1}{x} = 5 \), what is the value of \( 2x^2 + rac{2}{x^2} \)?

["Title: How to Solve ( x + \dfrac{1}{x} = 5 ) and Find the Value of ( 2x^2 + \dfrac{2}{x^2} ) – A Step-by-Step Guide", "Meta Description:\nSolve the equation ( x + \dfrac{1}{x} = 5 ) and compute ( 2x^2 + \dfrac{2}{x^2} ). Learn algebraic techniques to simplify expressions and unlock powerful math shortcuts.", "---", "### Introduction\nMathematics often presents problems that look simple at first but unlock deep algebraic skills. One classic example is the equation:", "[\nx + \dfrac{1}{x} = 5\n]\nWhile solving for ( x ) directly might feel straightforward, a more insightful approach explores how expressions like ( 2x^2 + \dfrac{2}{x^2} ) can be derived efficiently using algebraic identities. In this article, we’ll walk through the step-by-step solution, explain key formulas, and show how to compute the desired value with minimal effort.", "If you're studying algebra or preparing for competitive tests, mastering such techniques is invaluable. Let’s dive in!", "---", "### Step 1: Start with the Given Equation\nWe are given:\n[\nx + \dfrac{1}{x} = 5\n]\nThis expression is symmetric in ( x ) and ( \dfrac{1}{x} ), making it ideal for squaring both sides to generate higher powers.", "---", "### Step 2: Square Both Sides to Generate ( x^2 + \dfrac{1}{x^2} )\nTo find ( x^2 + \dfrac{1}{x^2} ), square both sides:\n[\n\left( x + \dfrac{1}{x} \right)^2 = 5^2 = 25\n]\nExpand the left-hand side:\n[\nx^2 + 2 \cdot x \cdot \dfrac{1}{x} + \dfrac{1}{x^2} = x^2 + 2 + \dfrac{1}{x^2}\n]\nThus:\n[\nx^2 + 2 + \dfrac{1}{x^2} = 25\n]\nSubtract 2 from both sides:\n[\nx^2 + \dfrac{1}{x^2} = 23\n]", "Key insight:\nInstead of solving for ( x ) explicitly (which leads to quadratic equations), squaring preserves the need to compute only the target expression ( 2x^2 + \dfrac{2}{x^2} = 2(x^2 + \dfrac{1}{x^2}) ), saving time and complexity.", "---", "### Step 3: Compute the Final Expression\nNow that we know:\n[\nx^2 + \dfrac{1}{x^2} = 23\n]\nMultiply both sides by 2:\n[\n2x^2 + \dfrac{2}{x^2} = 2 \cdot 23 = 46\n]", "---", "### Final Answer and Verification\n[\n\boxed{46}\n]\nSo, if ( x + \dfrac{1}{x} = 5 ), then:\n[\n2x^2 + \dfrac{2}{x^2} = 46\n]", "---", "### Why This Method Works\nThis approach leverages algebraic identities to bypass cumbersome root extraction and directly computes symmetric expressions. It’s especially powerful for olympiad-style problems, geometry, and calculus readiness, where simplifying expressions efficiently is crucial.", "---", "### Related Concepts\n- Arithmetic Mean–Geometric Mean (AM-GM) Inequality: For ( x > 0 ), ( x + \dfrac{1}{x} \geq 2 ), confirming our input is valid (since 5 > 2).\n- Quadratic Roots: Although not needed here, solving ( x + \dfrac{1}{x} = 5 ) leads to ( x^2 - 5x + 1 = 0 ), with solutions ( x = \dfrac{5 \pm \sqrt{21}}{2} ).\n- General Pattern: For ( x + \dfrac{1}{x} = k ), the identity ( x^n + \dfrac{1}{x^n} ) can be computed recursively.", "---", "### Conclusion\nUnderstanding how to manipulate expressions like ( x + \dfrac{1}{x} ) and apply squaring techniques unlocks elegant solutions to seemingly complex equations. Practice this method to strengthen your algebraic intuition—and remember: sometimes, the shortest path isn’t the direct one.", "Whether you’re prepping for exams, learning algebra, or building problem-solving skills, mastering identities like ( \left( x + \dfrac{1}{x} \right)^2 = x^2 + \dfrac{1}{x^2} + 2 ) is a foundational tool.", "Keywords: ( x + \dfrac{1}{x} ), ( 2x^2 + \dfrac{2}{x^2} ), algebraic identity, solve equations, simplify expressions, math trick, algebra practice, fraction equation, quadratic expression.", "---", "Have you encountered this type of problem before? Share your approach in the comments below!\nFor more algebraic techniques and practice problems, explore related math guides and start mastering your math skills today."]