Solution: Square both sides: \( x^2 + 2 + rac{1}{x^2} = 25 \), so \( x^2 + rac{1}{x^2} = 23 \). Multiply by 2: \( 2x^2 + rac{2}{x^2} = 46 \). oxed{46}

["Solution: Simplifying the Expression ( x^2 + 2 + \frac{1}{x^2} = 25 ) Step-by-Step", "When solving equations involving quadratic expressions like ( x^2 + \frac{1}{x^2} ), a useful first step is squaring both sides after isolating relevant terms. In this article, we explore how squaring the given equation leads to a clean simplification and reveals the key value ( 46 ).", "---", "### Understanding the Original Equation", "We begin with:", "[\nx^2 + 2 + \frac{1}{x^2} = 25\n]", "This equation contains ( x^2 + \frac{1}{x^2} ) and a constant term. To simplify, we isolate the quadratic expression:", "[\nx^2 + \frac{1}{x^2} = 25 - 2 = 23\n]", "Key insight: By subtracting 2 from both sides, we isolate the sum ( x^2 + \frac{1}{x^2} ), laying the foundation for further algebraic manipulation.", "---", "### Step 2: Multiply by 2 to Double the Expression", "Next, we multiply the entire equation by 2:", "[\n2 \left( x^2 + \frac{1}{x^2} \right) = 2 \ imes 23\n]", "[\n2x^2 + \frac{2}{x^2} = 46\n]", "Why multiply by 2? This step prepares the expression for easier factoring or manipulation later, especially when combining terms involving (x^2) and (\frac{1}{x^2}).", "---", "### The Completed Solution: Final Simplified Form", "After simplification, we arrive at:", "[\n2x^2 + \frac{2}{x^2} = 46\n]", "This form highlights a clean multiple of our target expression. The final boxed result is:", "[\n\boxed{46}\n]", "---", "### Why This Technique Works", "Squaring and multiplying identities like these are powerful in algebra because they transform asymmetric expressions into symmetric forms, often revealing deeper structures or simplifying the path to the solution. Though squaring wasn’t directly applied here in the strict definition, isolating and multiplying by 2 achieves a similar algebraic elegance.", "---", "### Takeaway for Students and Practitioners", "When solving equations with ( x^2 + \frac{1}{x^2} ), isolate that sum first, manipulate constants, and use multiplication by 2 (or similar coefficients) to uncover simplified values. These strategies save time and illuminate patterns crucial for solving higher-level problems.", "---", "SEO Keywords:\nSquare both sides algebraically, simplify ( x^2 + 2 + \frac{1}{x^2} = 25 ), multiply by 2, final expression = 46, solving quadratic expressions, algebraic manipulation, equation solving technique", "Meta Description:\nLearn how isolating ( x^2 + \frac{1}{x^2} ) and multiplying by 2 leads to the simplified result of 46 in this step-by-step algebraic solution. Perfect for students mastering quadratic identities and equation transformation."]