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

Options

A. `(62)/(27)`
B. `(52)/(27)`
C. `(61)/(27)`
D. `(8)/(27)`

Answer

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

### Step 1: Simplify the given equation
Starting with the 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
To eliminate the fraction, multiply both sides by \( x \):
\[ x(3x - 2) = 3 \]
This simplifies to:
\[ 3x^2 - 2x = 3 \]

### Step 3: Rearrange the equation
Rearranging gives us a standard quadratic equation:
\[ 3x^2 - 2x - 3 = 0 \]

### Step 4: Use the quadratic formula
We can solve for \( x \) using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 3, b = -2, c = -3 \):
\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 3 \cdot (-3)}}{2 \cdot 3} \]
Calculating the discriminant:
\[ x = \frac{2 \pm \sqrt{4 + 36}}{6} = \frac{2 \pm \sqrt{40}}{6} = \frac{2 \pm 2\sqrt{10}}{6} = \frac{1 \pm \sqrt{10}}{3} \]

### Step 5: Calculate \( x - \frac{1}{x} \)
Let \( y = x - \frac{1}{x} \). We can find \( y \) using:
\[ y = \frac{1 + \sqrt{10}}{3} - \frac{3}{1 + \sqrt{10}} \]
To simplify \( \frac{3}{1 + \sqrt{10}} \), multiply numerator and denominator by \( 1 - \sqrt{10} \):
\[ \frac{3(1 - \sqrt{10})}{(1 + \sqrt{10})(1 - \sqrt{10})} = \frac{3(1 - \sqrt{10})}{1 - 10} = \frac{3(1 - \sqrt{10})}{-9} = -\frac{1 - \sqrt{10}}{3} \]
Thus,
\[ y = \frac{1 + \sqrt{10}}{3} + \frac{1 - \sqrt{10}}{3} = \frac{2}{3} \]

### Step 6: Calculate \( x^3 - \frac{1}{x^3} \)
Using the identity:
\[ x^3 - \frac{1}{x^3} = (x - \frac{1}{x})\left((x - \frac{1}{x})^2 + 3\right) \]
Substituting \( y = \frac{2}{3} \):
\[ x^3 - \frac{1}{x^3} = \frac{2}{3}\left(\left(\frac{2}{3}\right)^2 + 3\right) = \frac{2}{3}\left(\frac{4}{9} + 3\right) = \frac{2}{3}\left(\frac{4}{9} + \frac{27}{9}\right) = \frac{2}{3}\left(\frac{31}{9}\right) = \frac{62}{27} \]

### Final Answer
Thus, the value of \( x^3 - \frac{1}{x^3} \) is:
\[ \frac{62}{27} \]