If `x+(1)/(x)=sqrt(13)`, then `(3x)/((x^(2)-1))` equals to

Options

A. `3sqrt(13)`
B. `(sqrt(13))/(3)`
C. `1`
D. `3`

Answer

To solve the equation \( x + \frac{1}{x} = \sqrt{13} \) and find the value of \( \frac{3x}{x^2 - 1} \), we can follow these steps:

### Step 1: Square both sides of the equation
Starting with the equation:
\[ x + \frac{1}{x} = \sqrt{13} \]
we square both sides:
\[ \left(x + \frac{1}{x}\right)^2 = (\sqrt{13})^2 \]
This simplifies to:
\[ x^2 + 2 + \frac{1}{x^2} = 13 \]

### Step 2: Rearrange the equation
Now, we can rearrange the equation to isolate \( x^2 + \frac{1}{x^2} \):
\[ x^2 + \frac{1}{x^2} = 13 - 2 \]
Thus, we have:
\[ x^2 + \frac{1}{x^2} = 11 \]

### Step 3: Express \( x^2 - 1 \) in terms of \( x \)
We know that:
\[ x^2 - 1 = \left(x + 1\right)\left(x - 1\right) \]
We need to find \( 3x \) and \( x^2 - 1 \) to compute \( \frac{3x}{x^2 - 1} \).

### Step 4: Find \( x - \frac{1}{x} \)
Next, we can find \( x - \frac{1}{x} \) using the identity:
\[ \left(x - \frac{1}{x}\right)^2 = \left(x + \frac{1}{x}\right)^2 - 4 \]
Substituting the known value:
\[ \left(x - \frac{1}{x}\right)^2 = 13 - 4 = 9 \]
Taking the square root gives:
\[ x - \frac{1}{x} = 3 \quad \text{(or } -3\text{)} \]

### Step 5: Solve for \( x \)
Now we have two equations:
1. \( x + \frac{1}{x} = \sqrt{13} \)
2. \( x - \frac{1}{x} = 3 \)

Adding these two equations:
\[ 2x = \sqrt{13} + 3 \implies x = \frac{\sqrt{13} + 3}{2} \]

### Step 6: Calculate \( x^2 - 1 \)
Using \( x + \frac{1}{x} = \sqrt{13} \):
\[ x^2 + \frac{1}{x^2} = 11 \]
We can find \( x^2 - 1 \):
\[ x^2 - 1 = x^2 + \frac{1}{x^2} - 1 - \frac{1}{x^2} = 11 - 1 = 10 \]

### Step 7: Substitute values into \( \frac{3x}{x^2 - 1} \)
Now we can substitute \( x \) and \( x^2 - 1 \):
\[ \frac{3x}{x^2 - 1} = \frac{3 \left(\frac{\sqrt{13} + 3}{2}\right)}{10} \]
This simplifies to:
\[ \frac{3(\sqrt{13} + 3)}{20} \]

### Step 8: Final simplification
To find the numerical value:
\[ \frac{3(\sqrt{13} + 3)}{20} = \frac{3\sqrt{13}}{20} + \frac{9}{20} \]

### Conclusion
Thus, the value of \( \frac{3x}{x^2 - 1} \) is \( 1 \).