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

Options

A. 1
B. 27
C. 0
D. `-27`

Answer

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

### Step 1: Eliminate the fractions
Multiply both sides of the equation by \( 3x \) (the least common multiple of the denominators) to eliminate the fractions:
\[ 3x \left( \frac{x}{3} \right) + 3x \left( \frac{3}{x} \right) = 3x \cdot 1 \]
This simplifies to:
\[ x^2 + 9 = 3x \]

### Step 2: Rearrange the equation
Rearranging the equation gives us:
\[ x^2 - 3x + 9 = 0 \]

### Step 3: Use the quadratic formula
To solve the quadratic equation \( x^2 - 3x + 9 = 0 \), we can apply the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -3 \), and \( c = 9 \). Plugging in these values:
\[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot 9}}{2 \cdot 1} \]
This simplifies to:
\[ x = \frac{3 \pm \sqrt{9 - 36}}{2} \]
\[ x = \frac{3 \pm \sqrt{-27}}{2} \]
Since \( \sqrt{-27} = 3i\sqrt{3} \), we have:
\[ x = \frac{3 \pm 3i\sqrt{3}}{2} \]

### Step 4: Find \( x^3 \)
Let \( x = \frac{3 + 3i\sqrt{3}}{2} \) or \( x = \frac{3 - 3i\sqrt{3}}{2} \). We can use the identity for the sum of cubes:
\[ x^3 + 27 = (x + 3)(x^2 - 3x + 9) \]
Since \( x^2 - 3x + 9 = 0 \), we have:
\[ x^3 + 27 = 0 \]
Thus,
\[ x^3 = -27 \]

### Final Answer
The value of \( x^3 \) is \( -27 \).

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