If `3x^2 - 9x + 3 = 0`, then what is the value of `(x + 1/x)^3` ?

Options

A. 9
B. 729
C. 81
D. 27

Answer

To solve the equation \(3x^2 - 9x + 3 = 0\) and find the value of \((x + \frac{1}{x})^3\), we can follow these steps:

### Step 1: Simplify the equation
Start with the given equation:
\[ 3x^2 - 9x + 3 = 0 \]
We can factor out a 3 from the equation:
\[ 3(x^2 - 3x + 1) = 0 \]
This simplifies to:
\[ x^2 - 3x + 1 = 0 \]

### Step 2: Solve the quadratic equation
Now we will use the quadratic formula to find the roots of the equation \(x^2 - 3x + 1 = 0\). The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 1\), \(b = -3\), and \(c = 1\). Plugging in these values:
\[ x = \frac{3 \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]
\[ x = \frac{3 \pm \sqrt{9 - 4}}{2} \]
\[ x = \frac{3 \pm \sqrt{5}}{2} \]

### Step 3: Find \(x + \frac{1}{x}\)
Next, we need to find \(x + \frac{1}{x}\). We can use the identity:
\[ x + \frac{1}{x} = \frac{x^2 + 1}{x} \]
Calculating \(x^2\) from our quadratic equation:
\[ x^2 = 3x - 1 \]
Thus:
\[ x + \frac{1}{x} = \frac{(3x - 1) + 1}{x} = \frac{3x}{x} = 3 \]

### Step 4: Calculate \((x + \frac{1}{x})^3\)
Now we can find \((x + \frac{1}{x})^3\):
\[ (x + \frac{1}{x})^3 = 3^3 = 27 \]

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