Expand: \( x^2 + 2 + \frac{1}{x^2} = 25 \)

["# Expand and Solve: A Comprehensive Guide to ( x^2 + 2 + \frac{1}{x^2} = 25 )", "Solving equations involving both ( x^2 ) and ( \frac{1}{x^2} \ can seem challenging at first, but with the right approach, it becomes manageable. This article explores how to expand and solve the equation ( x^2 + 2 + \frac{1}{x^2} = 25 ), offering step-by-step guidance and practical insights.", "## Step 1: Simplify and Reformulate the Equation", "Begin by rewriting the equación clearly:", "[\nx^2 + \frac{1}{x^2} + 2 = 25\n]", "Subtract 2 from both sides:", "[\nx^2 + \frac{1}{x^2} = 23\n]", "This form is easier to analyze, as it isolates the key expression ( x^2 + \frac{1}{x^2} ).", "## Step 2: Use a Useful Algebraic Identity", "Recall the identity:", "[\nx^2 + \frac{1}{x^2} = \left( x + \frac{1}{x} \right)^2 - 2\n]", "Applying this identity:", "[\n\left( x + \frac{1}{x} \right)^2 - 2 = 23\n]", "Add 2 to both sides:", "[\n\left( x + \frac{1}{x} \right)^2 = 25\n]", "Take square roots of both sides:", "[\nx + \frac{1}{x} = \pm 5\n]", "## Step 3: Solve the Two Resulting Linear Fractional Equations", "We now have two cases to solve:", "### Case 1: ( x + \frac{1}{x} = 5 )", "Multiply through by ( x ) (noting ( x <br/>\ne 0 )):", "[\nx^2 + 1 = 5x\n]", "Rearrange:", "[\nx^2 - 5x + 1 = 0\n]", "Use the quadratic formula:", "[\nx = \frac{5 \pm \sqrt{(-5)^2 - 4(1)(1)}}{2} = \frac{5 \pm \sqrt{25 - 4}}{2} = \frac{5 \pm \sqrt{21}}{2}\n]", "### Case 2: ( x + \frac{1}{x} = -5 )", "Similarly, multiply by ( x ):", "[\nx^2 + 1 = -5x\n]", "Rearrange:", "[\nx^2 + 5x + 1 = 0\n]", "Apply the quadratic formula:", "[\nx = \frac{-5 \pm \sqrt{25 - 4}}{2} = \frac{-5 \pm \sqrt{21}}{2}\n]", "## Step 4: Combine All Solutions", "Thus, the complete set of real solutions to the original equation is:", "[\nx = \frac{5 \pm \sqrt{21}}{2}, \quad x = \frac{-5 \pm \sqrt{21}}{2}, \quad \ ext{with } x <br/>\ne 0\n]", "All four values are valid, as none make the original expression undefined.", "## Why This Equation Matters — Real-World Relevance", "Equations of the form ( x^2 + \frac{1}{x^2} ) arise in:", "- Optimization problems where reciprocal quantities balance (e.g., ratios in physics or economics).\n- Signal processing and control theory, where signals are analyzed in terms of amplitude and inverse amplitude.\n- Advanced algebra and number theory, as a foundation for solving symmetric rational expressions.", "## SEO Keywords and Phrases to Boost Visibility:", "- Expand and solve ( x^2 + \frac{1}{x^2} = 25 )\n- Solve rational algebraic equation step-by-step\n- How to solve ( x^2 + 2 + \frac{1}{x^2} = 25 )\n- Real solutions to ( x^2 + 1/x^2 = 23 )\n- Algebraic identities for ( x + 1/x )\n- Quadratic equation from reciprocal expressions\n- Step-by-step polynomial solution guide", "## Conclusion", "Expanding and solving equations involving ( x^2 ) and ( \frac{1}{x^2} ) relies on recognizing key algebraic identities and reducing higher-degree expressions into manageable quadratics. Mastering these steps enables solve complex symmetric rational equations with confidence and precision.", "For further practice, try similar equations like ( x^2 + \frac{9}{x^2} = 32 ), or explore how substitutions simplify equations involving reciprocal terms.", "---", "If you're studying algebra or preparing for advanced math exams, refining techniques like this will strengthen your problem-solving toolkit and improve your ability to tackle symmetric rational equations with ease."]