If `4x^(2) - 6x + 1 = 0`, then the value of `8x^(3) + (8x^(3))^(-1)` is :

Options

A. 36
B. 13
C. 18
D. 11

Answer

To solve the equation \(4x^2 - 6x + 1 = 0\) and find the value of \(8x^3 + (8x^3)^{-1}\), we will follow these steps:

### Step 1: Simplify the given equation
We start with the equation:
\[ 4x^2 - 6x + 1 = 0 \]
We can divide the entire equation by \(2x\) (assuming \(x \neq 0\)):
\[ \frac{4x^2}{2x} - \frac{6x}{2x} + \frac{1}{2x} = 0 \]
This simplifies to:
\[ 2x - 3 + \frac{1}{2x} = 0 \]

### Step 2: Rearrange the equation
Now, we can rearrange this to isolate the term involving \(x\):
\[ 2x + \frac{1}{2x} = 3 \]

### Step 3: Cube both sides
Next, we cube both sides of the equation:
\[ \left(2x + \frac{1}{2x}\right)^3 = 3^3 \]
This gives us:
\[ \left(2x + \frac{1}{2x}\right)^3 = 27 \]

### Step 4: Apply the cube expansion formula
Using the formula for the cube of a binomial, \( (a + b)^3 = a^3 + b^3 + 3ab(a + b) \), where \(a = 2x\) and \(b = \frac{1}{2x}\):
\[ (2x)^3 + \left(\frac{1}{2x}\right)^3 + 3(2x)\left(\frac{1}{2x}\right)(2x + \frac{1}{2x}) = 27 \]
Calculating each term:
\[ 8x^3 + \frac{1}{8x^3} + 3(1)(3) = 27 \]

### Step 5: Simplify the equation
This simplifies to:
\[ 8x^3 + \frac{1}{8x^3} + 9 = 27 \]
Now, we can isolate \(8x^3 + \frac{1}{8x^3}\):
\[ 8x^3 + \frac{1}{8x^3} = 27 - 9 \]
\[ 8x^3 + \frac{1}{8x^3} = 18 \]

### Final Answer
Thus, the value of \(8x^3 + (8x^3)^{-1}\) is:
\[ \boxed{18} \]