If `x (3- (2)/(x)) = (3)/(x)` then the value of `x ^(2) + (1)/(x ^(2))` will be

Options

A. `3 (1)/(9)`
B. `3 (2)/(9)`
C. `2 (1)/(9)`
D. `2 (4)/(9)`

Answer

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

### Step 1: Simplify the Equation
Start by simplifying the left-hand side of the equation:
\[ x(3 - \frac{2}{x}) = 3x - 2 \]
Thus, the equation becomes:
\[ 3x - 2 = \frac{3}{x} \]

### Step 2: Eliminate the Fraction
To eliminate the fraction, multiply both sides of the equation by \( x \):
\[ x(3x - 2) = 3 \]
This simplifies to:
\[ 3x^2 - 2x - 3 = 0 \]

### Step 3: Solve the Quadratic Equation
Now, we will solve the quadratic equation \( 3x^2 - 2x - 3 = 0 \) using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 3 \), \( b = -2 \), and \( c = -3 \). Plugging in these values:
\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 3 \cdot (-3)}}{2 \cdot 3} \]
This simplifies to:
\[ x = \frac{2 \pm \sqrt{4 + 36}}{6} = \frac{2 \pm \sqrt{40}}{6} = \frac{2 \pm 2\sqrt{10}}{6} = \frac{1 \pm \sqrt{10}}{3} \]

### Step 4: Find \( x^2 + \frac{1}{x^2} \)
Next, we need to find \( x^2 + \frac{1}{x^2} \). We can use the identity:
\[ x^2 + \frac{1}{x^2} = \left(x - \frac{1}{x}\right)^2 + 2 \]
First, we need to find \( x - \frac{1}{x} \). We can calculate \( x - \frac{1}{x} \) using the values of \( x \):
\[ x - \frac{1}{x} = \frac{1 + \sqrt{10}}{3} - \frac{3}{1 + \sqrt{10}} = \frac{(1 + \sqrt{10})^2 - 3}{3(1 + \sqrt{10})} \]
Calculating \( (1 + \sqrt{10})^2 - 3 \):
\[ = 1 + 2\sqrt{10} + 10 - 3 = 8 + 2\sqrt{10} \]
Thus:
\[ x - \frac{1}{x} = \frac{8 + 2\sqrt{10}}{3(1 + \sqrt{10})} \]

### Step 5: Calculate \( x^2 + \frac{1}{x^2} \)
Now we can substitute back into the identity:
\[ x^2 + \frac{1}{x^2} = \left(\frac{8 + 2\sqrt{10}}{3(1 + \sqrt{10})}\right)^2 + 2 \]
After simplifying, we find:
\[ x^2 + \frac{1}{x^2} = \frac{64 + 32\sqrt{10} + 40}{9(1 + 10 + 2\sqrt{10})} + 2 \]
This will yield a final numerical value.

### Final Answer
After performing the calculations, we find that:
\[ x^2 + \frac{1}{x^2} = \frac{64 + 32\sqrt{10} + 40 + 18}{9} = \frac{122 + 32\sqrt{10}}{9} \]
However, the specific numerical value will depend on the simplification.