If `x^(4)+x^(-4)=2599`, then one of the values of `x-x^(-1)`, where `xlt0` is equal to:

Options

A. 9
B. 7
C. 5
D. 8

Answer

To solve the equation \( x^4 + x^{-4} = 2599 \) and find the value of \( x - x^{-1} \) where \( x < 0 \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ x^4 + x^{-4} = 2599 \]

### Step 2: Use the identity for squares
We can use the identity:
\[ x^4 + x^{-4} = (x^2 + x^{-2})^2 - 2 \]
This allows us to express \( x^4 + x^{-4} \) in terms of \( x^2 + x^{-2} \).

### Step 3: Set up the equation
Substituting into our equation gives us:
\[ (x^2 + x^{-2})^2 - 2 = 2599 \]
Adding 2 to both sides:
\[ (x^2 + x^{-2})^2 = 2601 \]

### Step 4: Take the square root
Taking the square root of both sides, we find:
\[ x^2 + x^{-2} = \sqrt{2601} \]
Calculating the square root:
\[ \sqrt{2601} = 51 \]
So we have:
\[ x^2 + x^{-2} = 51 \]

### Step 5: Use another identity
We can use another identity:
\[ x^2 + x^{-2} = (x - x^{-1})^2 + 2 \]
Substituting this into our equation gives:
\[ (x - x^{-1})^2 + 2 = 51 \]

### Step 6: Solve for \( (x - x^{-1})^2 \)
Subtracting 2 from both sides:
\[ (x - x^{-1})^2 = 49 \]

### Step 7: Take the square root again
Taking the square root of both sides:
\[ x - x^{-1} = \pm 7 \]

### Step 8: Determine the correct sign
Since we are given that \( x < 0 \), we choose the negative value:
\[ x - x^{-1} = -7 \]

### Conclusion
Thus, one of the values of \( x - x^{-1} \) where \( x < 0 \) is:
\[ \boxed{-7} \]