If ` 6 x ^(2) - 12 x + 1 = 0 ` then the value of ` 27 x^(3) + (1)/( 8 x^(3))` is

Options

A. 162
B. 189
C. 207
D. 225

Answer

To solve the equation \( 6x^2 - 12x + 1 = 0 \) and find the value of \( 27x^3 + \frac{1}{8x^3} \), we can follow these steps:

### Step 1: Simplify the original equation
We start with the equation:
\[ 6x^2 - 12x + 1 = 0 \]
We can divide the entire equation by 2 to simplify it:
\[ 3x^2 - 6x + \frac{1}{2} = 0 \]

### Step 2: Express \( 3x + \frac{1}{2x} \)
We can rearrange the equation to express \( 3x + \frac{1}{2x} \):
\[ 3x + \frac{1}{2x} = 6 \]

### Step 3: Identify \( a \) and \( b \)
Let:
\[ a = 3x \quad \text{and} \quad b = \frac{1}{2x} \]
Then, we need to find \( a^3 + b^3 \).

### Step 4: Use the identity for cubes
We can use the identity:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
First, we calculate \( a + b \):
\[ a + b = 3x + \frac{1}{2x} = 6 \]

### Step 5: Calculate \( ab \)
Next, we calculate \( ab \):
\[ ab = 3x \cdot \frac{1}{2x} = \frac{3}{2} \]

### Step 6: Calculate \( a^2 + b^2 \)
Now, we need \( a^2 + b^2 \):
\[ a^2 + b^2 = (a + b)^2 - 2ab = 6^2 - 2 \cdot \frac{3}{2} = 36 - 3 = 33 \]

### Step 7: Substitute into the identity
Now we can substitute into the identity:
\[ a^3 + b^3 = (a + b)((a^2 + b^2) - ab) = 6(33 - \frac{3}{2}) = 6 \left( \frac{66 - 3}{2} \right) = 6 \cdot \frac{63}{2} = \frac{378}{2} = 189 \]

### Conclusion
Thus, the value of \( 27x^3 + \frac{1}{8x^3} \) is:
\[ \boxed{189} \]