If `2[x^(2)+(1)/(x^(2))]-2[x-(1)/(x)]-8=0`, what are the two values of `(x-(1)/(x))`?

Options

A. `-1or2`
B. `1or-2`
C. `-1or-2`
D. `1or2`

Answer

To solve the equation \(2[x^2 + \frac{1}{x^2}] - 2[x - \frac{1}{x}] - 8 = 0\), we will follow these steps:

### Step 1: Rewrite the equation
The given equation can be rewritten as:
\[ 2\left(x^2 + \frac{1}{x^2}\right) - 2\left(x - \frac{1}{x}\right) - 8 = 0 \]

### Step 2: Substitute \(k = x - \frac{1}{x}\)
Let \(k = x - \frac{1}{x}\). We know that:
\[ x^2 + \frac{1}{x^2} = (x - \frac{1}{x})^2 + 2 = k^2 + 2 \]

### Step 3: Substitute into the equation
Substituting \(x^2 + \frac{1}{x^2}\) in the equation gives:
\[ 2(k^2 + 2) - 2k - 8 = 0 \]

### Step 4: Simplify the equation
Expanding and simplifying:
\[ 2k^2 + 4 - 2k - 8 = 0 \]
\[ 2k^2 - 2k - 4 = 0 \]

### Step 5: Divide the entire equation by 2
Dividing the equation by 2 to simplify:
\[ k^2 - k - 2 = 0 \]

### Step 6: Factor the quadratic equation
Now we will factor the quadratic equation:
\[ (k - 2)(k + 1) = 0 \]

### Step 7: Solve for \(k\)
Setting each factor to zero gives us:
\[ k - 2 = 0 \quad \Rightarrow \quad k = 2 \]
\[ k + 1 = 0 \quad \Rightarrow \quad k = -1 \]

### Step 8: Conclusion
The two values of \(k\) (which is \(x - \frac{1}{x}\)) are:
\[ k = 2 \quad \text{and} \quad k = -1 \]