If `x ^(2) - 3 sqrt2 x +1 = 0,` then the value of `x ^(3) + (1)/(x ^(3))` is :

Options

A. A)`45 sqrt2`
B. B)`54 sqrt2`
C. C)`24 sqrt6`
D. D)`36 sqrt6`

Answer

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

### Step 1: Solve the quadratic equation
We start with the quadratic equation:
\[ x^2 - 3\sqrt{2}x + 1 = 0 \]
We can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1 \), \( b = -3\sqrt{2} \), and \( c = 1 \).

### Step 2: Calculate the discriminant
First, we calculate the discriminant \( b^2 - 4ac \):
\[ b^2 = (3\sqrt{2})^2 = 18 \]
\[ 4ac = 4 \cdot 1 \cdot 1 = 4 \]
Thus, the discriminant is:
\[ b^2 - 4ac = 18 - 4 = 14 \]

### Step 3: Find the roots
Now we can substitute back into the quadratic formula:
\[ x = \frac{3\sqrt{2} \pm \sqrt{14}}{2} \]

### Step 4: Calculate \( x + \frac{1}{x} \)
Next, we need to find \( x + \frac{1}{x} \). We can use the identity:
\[ x + \frac{1}{x} = \frac{x^2 + 1}{x} \]
From the quadratic equation, we know:
\[ x^2 = 3\sqrt{2}x - 1 \]
Thus:
\[ x + \frac{1}{x} = \frac{(3\sqrt{2}x - 1) + 1}{x} = 3\sqrt{2} - \frac{1}{x} \]
To find \( \frac{1}{x} \), we can use the roots we calculated.

### Step 5: Calculate \( x^3 + \frac{1}{x^3} \)
Using the identity:
\[ x^3 + \frac{1}{x^3} = \left(x + \frac{1}{x}\right)^3 - 3\left(x + \frac{1}{x}\right) \]
Let \( y = x + \frac{1}{x} \). We can find \( y \) from \( 3\sqrt{2} \).

### Step 6: Substitute and simplify
Substituting \( y = 3\sqrt{2} \):
\[ x^3 + \frac{1}{x^3} = (3\sqrt{2})^3 - 3(3\sqrt{2}) \]
Calculating \( (3\sqrt{2})^3 \):
\[ (3\sqrt{2})^3 = 27 \cdot 2\sqrt{2} = 54\sqrt{2} \]
Now substituting back:
\[ x^3 + \frac{1}{x^3} = 54\sqrt{2} - 9\sqrt{2} = 45\sqrt{2} \]

### Final Result
Thus, the value of \( x^3 + \frac{1}{x^3} \) is:
\[ \boxed{45\sqrt{2}} \]