If ` 3x^2 - 4x - 3=0` then the value of ` x- (1)/(x)` is

Options

A. 3
B. 4
C. `3/4`
D. `4/3`

Answer

To solve the equation \( 3x^2 - 4x - 3 = 0 \) and find the value of \( x - \frac{1}{x} \), we can follow these steps:

### Step 1: Solve the quadratic equation
We start with the quadratic equation:
\[ 3x^2 - 4x - 3 = 0 \]
We can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 3 \), \( b = -4 \), and \( c = -3 \).

### Step 2: Calculate the discriminant
First, we calculate the discriminant \( b^2 - 4ac \):
\[ b^2 - 4ac = (-4)^2 - 4 \cdot 3 \cdot (-3) = 16 + 36 = 52 \]

### Step 3: Substitute into the quadratic formula
Now we substitute into the quadratic formula:
\[ x = \frac{-(-4) \pm \sqrt{52}}{2 \cdot 3} = \frac{4 \pm \sqrt{52}}{6} \]
We can simplify \( \sqrt{52} \) to \( 2\sqrt{13} \):
\[ x = \frac{4 \pm 2\sqrt{13}}{6} = \frac{2 \pm \sqrt{13}}{3} \]

### Step 4: Find \( x - \frac{1}{x} \)
Next, we need to find \( x - \frac{1}{x} \). We can start by calculating \( \frac{1}{x} \):
\[ x = \frac{2 + \sqrt{13}}{3} \quad \text{(taking the positive root for simplicity)} \]
Thus,
\[ \frac{1}{x} = \frac{3}{2 + \sqrt{13}} \]
To rationalize the denominator:
\[ \frac{1}{x} = \frac{3(2 - \sqrt{13})}{(2 + \sqrt{13})(2 - \sqrt{13})} = \frac{3(2 - \sqrt{13})}{4 - 13} = \frac{3(2 - \sqrt{13})}{-9} = -\frac{1}{3}(2 - \sqrt{13}) = -\frac{2}{3} + \frac{\sqrt{13}}{9} \]

### Step 5: Calculate \( x - \frac{1}{x} \)
Now we can calculate \( x - \frac{1}{x} \):
\[ x - \frac{1}{x} = \frac{2 + \sqrt{13}}{3} + \left(\frac{2}{3} - \frac{\sqrt{13}}{9}\right) \]
Combining the terms:
\[ x - \frac{1}{x} = \frac{2 + \sqrt{13} + 2 - \frac{\sqrt{13}}{3}}{3} \]
This simplifies to:
\[ = \frac{4 + \frac{3\sqrt{13} - \sqrt{13}}{3}}{3} = \frac{4 + \frac{2\sqrt{13}}{3}}{3} \]

### Step 6: Final simplification
To find \( x - \frac{1}{x} \), we can also use the relation derived from the quadratic equation:
\[ 3\left(x - \frac{1}{x}\right) = 4 \implies x - \frac{1}{x} = \frac{4}{3} \]

Thus, the value of \( x - \frac{1}{x} \) is:
\[ \boxed{\frac{4}{3}} \]