Solving Algebraic Expression Value

We are given an algebraic equation and asked to find the value of a specific algebraic expression based on this equation. The given equation is \( 2x - \frac{1}{2x} = 5 \), where \( x \neq 0 \). We need to find the value of \( x^{2} + \frac{1}{16x^{2} } - 2 \).

Step-by-Step Solution to Find the Value

To find the value of the expression \( x^{2} + \frac{1}{16x^{2} } - 2 \), we should use the given equation \( 2x - \frac{1}{2x} = 5 \) and manipulate it. Our goal is to create terms like \( x^2 \) and \( \frac{1}{16x^2} \) from the given equation.

1. Start with the given equation:
   The initial equation provided is \( 2x - \frac{1}{2x} = 5 \).
2. Modify the equation:
   We notice the target expression has \( x^2 \) and \( \frac{1}{16x^2} \). If we square the original equation \( 2x - \frac{1}{2x} = 5 \), we would get \( (2x)^2 = 4x^2 \) and \( (\frac{1}{2x})^2 = \frac{1}{4x^2} \). To get terms more directly related to \( x^2 \) and \( \frac{1}{16x^2} \), let's divide the entire original equation by 2:
   \( \frac{1}{2} \left( 2x - \frac{1}{2x} \right) = \frac{5}{2} \)
   This simplifies to:
   \( x - \frac{1}{4x} = \frac{5}{2} \)
3. Square both sides of the modified equation:
   Now, squaring both sides of the equation \( x - \frac{1}{4x} = \frac{5}{2} \) will introduce the square terms we are looking for:
   \( \left( x - \frac{1}{4x} \right)^2 = \left( \frac{5}{2} \right)^2 \)
4. Expand and simplify the squared equation:
   Using the algebraic identity for the square of a difference, \( (a-b)^2 = a^2 - 2ab + b^2 \), with \( a=x \) and \( b=\frac{1}{4x} \):
   \( x^2 - 2(x)\left(\frac{1}{4x}\right) + \left(\frac{1}{4x}\right)^2 = \frac{5^2}{2^2} \)
   Simplify the terms:
   \( x^2 - 2 \cdot \frac{x}{4x} + \frac{1^2}{(4x)^2} = \frac{25}{4} \)
   \( x^2 - \frac{2}{4} + \frac{1}{16x^2} = \frac{25}{4} \)
   \( x^2 - \frac{1}{2} + \frac{1}{16x^2} = \frac{25}{4} \)
5. Rearrange to find the value of \( x^2 + \frac{1}{16x^2} \):
   Let's isolate the terms \( x^2 + \frac{1}{16x^2} \) by moving the constant term \( -\frac{1}{2} \) to the right side of the equation:
   \( x^2 + \frac{1}{16x^2} = \frac{25}{4} + \frac{1}{2} \)
   To add the fractions on the right side, find a common denominator, which is 4:
   \( x^2 + \frac{1}{16x^2} = \frac{25}{4} + \frac{1 \cdot 2}{2 \cdot 2} \)
   \( x^2 + \frac{1}{16x^2} = \frac{25}{4} + \frac{2}{4} \)
   Combine the fractions:
   \( x^2 + \frac{1}{16x^2} = \frac{25 + 2}{4} \)
   \( x^2 + \frac{1}{16x^2} = \frac{27}{4} \)
6. Substitute the value into the target expression:
   The expression we were asked to evaluate is \( x^{2} + \frac{1}{16x^{2} } - 2 \). We have just found that \( x^{2} + \frac{1}{16x^{2} } = \frac{27}{4} \). Substitute this value into the expression:
   Value \( = \left( x^{2} + \frac{1}{16x^{2} } \right) - 2 \)
   Value \( = \frac{27}{4} - 2 \)
   To perform the subtraction, write 2 as a fraction with denominator 4:
   Value \( = \frac{27}{4} - \frac{2 \cdot 4}{1 \cdot 4} \)
   Value \( = \frac{27}{4} - \frac{8}{4} \)
   Perform the subtraction of the fractions:
   Value \( = \frac{27 - 8}{4} \)
   Value \( = \frac{19}{4} \)

Final Result

Based on the steps above, the value of the expression \( x^{2} + \frac{1}{16x^{2} } - 2 \) is \( \frac{19}{4} \).

Revision Table: Key Concepts in Algebraic Manipulation

This problem involves manipulating equations and using algebraic identities.

Additional Information: Solving Problems with Related Expressions

When you are given an equation involving a variable and asked to find the value of an expression involving related terms (like squares of the variable or its reciprocal), consider these approaches:

• Look for ways to square (or cube, etc.) the given equation or a modified version of it to generate the required terms.
• Use algebraic identities like \( (a+b)^2 \), \( (a-b)^2 \), \( a^2-b^2 \), etc., to simplify or transform expressions.
• Sometimes you might need to find the value of intermediate expressions first (like finding \( x^2 + \frac{1}{16x^2} \) in this case) before substituting into the final expression.
• Always check the constraints on the variables (like \( x \neq 0 \)) to ensure expressions are defined.
• Practice recognizing patterns between the given equation and the target expression to choose the right manipulation strategy.