If `x-3/x=6,x ne 0`, then the value of `(x^4-(27)/(x^2))/(x^2-3x-3)` is :

Options

A. 54
B. 270
C. 80
D. 90

Answer

To solve the equation given \( \frac{x - 3}{x} = 6 \) and find the value of \( \frac{x^4 - \frac{27}{x^2}}{x^2 - 3x - 3} \), we will follow these steps:

### Step 1: Solve for \( x \)
Starting with the equation:
\[ \frac{x - 3}{x} = 6 \]
We can multiply both sides by \( x \) (since \( x \neq 0 \)):
\[ x - 3 = 6x \]
Rearranging gives:
\[ x - 6x = 3 \implies -5x = 3 \implies x = -\frac{3}{5} \]

### Step 2: Substitute \( x \) into the expression
Now we substitute \( x = -\frac{3}{5} \) into the expression \( \frac{x^4 - \frac{27}{x^2}}{x^2 - 3x - 3} \).

### Step 3: Calculate \( x^4 \)
First, we calculate \( x^4 \):
\[ x^4 = \left(-\frac{3}{5}\right)^4 = \frac{81}{625} \]

### Step 4: Calculate \( x^2 \)
Next, we calculate \( x^2 \):
\[ x^2 = \left(-\frac{3}{5}\right)^2 = \frac{9}{25} \]

### Step 5: Calculate \( \frac{27}{x^2} \)
Now, we calculate \( \frac{27}{x^2} \):
\[ \frac{27}{x^2} = \frac{27}{\frac{9}{25}} = 27 \cdot \frac{25}{9} = \frac{675}{9} = 75 \]

### Step 6: Substitute into the numerator
Now, we substitute into the numerator:
\[ x^4 - \frac{27}{x^2} = \frac{81}{625} - 75 = \frac{81}{625} - \frac{46875}{625} = \frac{81 - 46875}{625} = \frac{-46894}{625} \]

### Step 7: Calculate the denominator
Now, we calculate the denominator \( x^2 - 3x - 3 \):
\[ x^2 - 3x - 3 = \frac{9}{25} - 3\left(-\frac{3}{5}\right) - 3 \]
Calculating \( -3\left(-\frac{3}{5}\right) = \frac{9}{5} \):
\[ = \frac{9}{25} + \frac{45}{25} - \frac{75}{25} = \frac{9 + 45 - 75}{25} = \frac{-21}{25} \]

### Step 8: Combine the results
Now we can combine the results:
\[ \frac{x^4 - \frac{27}{x^2}}{x^2 - 3x - 3} = \frac{\frac{-46894}{625}}{\frac{-21}{25}} = \frac{-46894 \cdot 25}{-21 \cdot 625} = \frac{46894 \cdot 25}{21 \cdot 625} \]

### Step 9: Simplify the expression
Now we simplify:
\[ = \frac{46894 \cdot 25}{21 \cdot 625} = \frac{46894}{21 \cdot 25} = \frac{46894}{525} \]

### Final Answer
Thus, the value of the expression is:
\[ \frac{46894}{525} \]