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

Options

A. `2 (1)/(9)`
B. `2 (4)/(9)`
C. `3(1)/(9)`
D. `3 (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 with the given equation:
\[ x(3 - \frac{2}{x}) = \frac{3}{x} \]
Distributing \( x \) on the left side:
\[ 3x - 2 = \frac{3}{x} \]

### Step 2: Eliminate the fraction
Multiply both sides by \( x \) to eliminate the fraction:
\[ x(3x - 2) = 3 \]
This simplifies to:
\[ 3x^2 - 2x - 3 = 0 \]

### Step 3: Solve the quadratic equation
Now, we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 3 \), \( b = -2 \), and \( c = -3 \):
\[ b^2 - 4ac = (-2)^2 - 4(3)(-3) = 4 + 36 = 40 \]
Thus, the solutions for \( x \) are:
\[ x = \frac{2 \pm \sqrt{40}}{6} = \frac{2 \pm 2\sqrt{10}}{6} = \frac{1 \pm \sqrt{10}}{3} \]

### Step 4: Find \( x + \frac{1}{x} \)
To find \( x^2 + \frac{1}{x^2} \), we first need \( x + \frac{1}{x} \). We can compute \( \frac{1}{x} \) for both solutions:
\[ x = \frac{1 + \sqrt{10}}{3} \implies \frac{1}{x} = \frac{3}{1 + \sqrt{10}} \cdot \frac{1 - \sqrt{10}}{1 - \sqrt{10}} = \frac{3(1 - \sqrt{10})}{-9} = \frac{-1 + \frac{\sqrt{10}}{3}}{3} \]
Calculating \( x + \frac{1}{x} \):
\[ x + \frac{1}{x} = \frac{1 + \sqrt{10}}{3} + \frac{-1 + \sqrt{10}}{3} = \frac{2\sqrt{10}}{3} \]

### Step 5: Calculate \( x^2 + \frac{1}{x^2} \)
Using the identity:
\[ x^2 + \frac{1}{x^2} = \left(x + \frac{1}{x}\right)^2 - 2 \]
Substituting in the value we found:
\[ x^2 + \frac{1}{x^2} = \left(\frac{2\sqrt{10}}{3}\right)^2 - 2 = \frac{40}{9} - 2 = \frac{40}{9} - \frac{18}{9} = \frac{22}{9} \]

Thus, the final value of \( x^2 + \frac{1}{x^2} \) is:
\[ \boxed{\frac{22}{9}} \]