If `2x = 3 +(2)/(x) , ` then the value of `(4x)/(x ^(2) -1 x -1)` is :

Options

A. 4
B. 2
C. `8/5`
D. 8

Answer

To solve the equation \( 2x = 3 + \frac{2}{x} \) and find the value of \( \frac{4x}{x^2 - 1x - 1} \), we will follow these steps:

### Step 1: Solve for \( x \)
Starting with the equation:
\[ 2x = 3 + \frac{2}{x} \]
We can multiply both sides by \( x \) to eliminate the fraction:
\[ 2x^2 = 3x + 2 \]
Rearranging this gives us:
\[ 2x^2 - 3x - 2 = 0 \]

### Step 2: Factor the quadratic equation
Next, we will factor the quadratic equation \( 2x^2 - 3x - 2 = 0 \). We need two numbers that multiply to \( 2 \times -2 = -4 \) and add to \( -3 \). The numbers \( -4 \) and \( 1 \) work:
\[ 2x^2 - 4x + x - 2 = 0 \]
Grouping the terms:
\[ (2x^2 - 4x) + (x - 2) = 0 \]
Factoring by grouping:
\[ 2x(x - 2) + 1(x - 2) = 0 \]
This gives us:
\[ (2x + 1)(x - 2) = 0 \]

### Step 3: Find the values of \( x \)
Setting each factor to zero gives:
\[ 2x + 1 = 0 \quad \Rightarrow \quad x = -\frac{1}{2} \]
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]

### Step 4: Substitute \( x \) into the expression
Now we need to find the value of \( \frac{4x}{x^2 - 1x - 1} \). We will evaluate this for both values of \( x \).

#### For \( x = 2 \):
Calculating the denominator:
\[ x^2 - 1x - 1 = 2^2 - 2 - 1 = 4 - 2 - 1 = 1 \]
Calculating the expression:
\[ \frac{4(2)}{1} = \frac{8}{1} = 8 \]

#### For \( x = -\frac{1}{2} \):
Calculating the denominator:
\[ x^2 - 1x - 1 = \left(-\frac{1}{2}\right)^2 - \left(-\frac{1}{2}\right) - 1 = \frac{1}{4} + \frac{1}{2} - 1 = \frac{1}{4} + \frac{2}{4} - \frac{4}{4} = -\frac{1}{4} \]
Calculating the expression:
\[ \frac{4\left(-\frac{1}{2}\right)}{-\frac{1}{4}} = \frac{-2}{-\frac{1}{4}} = 2 \times 4 = 8 \]

### Conclusion
In both cases, the value of \( \frac{4x}{x^2 - 1x - 1} \) is \( 8 \).

### Final Answer
The value of \( \frac{4x}{x^2 - 1x - 1} \) is \( 8 \).