If `3x-(4)/(x)=4`, find `27x^(3)-(64)/(x^(3))`

Options

A. `28`
B. `208`
C. `138`
D. `200`

Answer

To solve the equation \( 3x - \frac{4}{x} = 4 \) and find the value of \( 27x^3 - \frac{64}{x^3} \), we can follow these steps:

### Step 1: Solve for \( 3x - \frac{4}{x} \)
We start with the equation given:
\[ 3x - \frac{4}{x} = 4 \]

### Step 2: Isolate \( 3x \)
Add \( \frac{4}{x} \) to both sides:
\[ 3x = 4 + \frac{4}{x} \]

### Step 3: Multiply through by \( x \)
To eliminate the fraction, multiply both sides by \( x \):
\[ 3x^2 = 4x + 4 \]

### Step 4: Rearrange the equation
Rearranging gives us a standard quadratic equation:
\[ 3x^2 - 4x - 4 = 0 \]

### Step 5: Use the quadratic formula
We can solve for \( x \) using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 3, b = -4, c = -4 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-4)^2 - 4 \cdot 3 \cdot (-4) = 16 + 48 = 64 \]
Now substitute into the formula:
\[ x = \frac{4 \pm \sqrt{64}}{2 \cdot 3} = \frac{4 \pm 8}{6} \]
This gives us two possible values for \( x \):
1. \( x = \frac{12}{6} = 2 \)
2. \( x = \frac{-4}{6} = -\frac{2}{3} \)

### Step 6: Calculate \( 27x^3 - \frac{64}{x^3} \)
We will calculate \( 27x^3 - \frac{64}{x^3} \) for both values of \( x \).

#### For \( x = 2 \):
\[ 27(2^3) - \frac{64}{(2^3)} = 27(8) - \frac{64}{8} = 216 - 8 = 208 \]

#### For \( x = -\frac{2}{3} \):
Calculating \( x^3 \):
\[ x^3 = \left(-\frac{2}{3}\right)^3 = -\frac{8}{27} \]
Now substituting:
\[ 27\left(-\frac{8}{27}\right) - \frac{64}{-\frac{8}{27}} = -8 + 8 = 0 \]

### Conclusion
The value of \( 27x^3 - \frac{64}{x^3} \) for \( x = 2 \) is \( 208 \) and for \( x = -\frac{2}{3} \) is \( 0 \). Since the problem asks for the value based on the original equation, we take \( x = 2 \).

Thus, the final answer is:
\[ \boxed{208} \]