Question

If \(x - \frac{1}{x} = 2\), then what is the value of \(x^2 + \frac{1}{x^2}\)?

• A. 4
• B. 5
• C. 3
• D. 6

Hint:

Squaring expressions of the form \(x \pm \frac{1}{x}\) is a common technique to find higher powers.

Answer 1

The Correct Option is D

To find the value of \( x^2 + \frac{1}{x^2} \) given that \( x - \frac{1}{x} = 2 \), we can proceed with the following steps:

1. First, square the given equation \( x - \frac{1}{x} = 2 \) to establish a relationship that includes \( x^2 \) and \( \frac{1}{x^2} \): \[ (x - \frac{1}{x})^2 = 2^2 \]
   \[ x^2 - 2x \cdot \frac{1}{x} + \frac{1}{x^2} = 4 \]
2. Since \( 2x \cdot \frac{1}{x} = 2 \), this can be simplified to: \[ x^2 - 2 + \frac{1}{x^2} = 4 \]
3. Rearrange the equation to solve for \( x^2 + \frac{1}{x^2} \): \[ x^2 + \frac{1}{x^2} = 4 + 2 = 6 \]

Thus, the value of \( x^2 + \frac{1}{x^2} \) is \( 6 \). Therefore, the correct answer is 6.