If `2x - 3//x = 2`, then find the value of `16x^4 + 81//x^4. `

Options

A. 162
B. 184
C. 206
D. 192

Answer

To solve the equation \( \frac{2x - 3}{x} = 2 \) and find the value of \( 16x^4 + \frac{81}{x^4} \), we can follow these steps:

### Step 1: Solve the initial equation
Starting with the equation:
\[ \frac{2x - 3}{x} = 2 \]
Multiply both sides by \( x \) (assuming \( x \neq 0 \)):
\[ 2x - 3 = 2x \]
This simplifies to:
\[ 2x - 3 = 2x \]
Subtract \( 2x \) from both sides:
\[ -3 = 0 \]
This indicates that we need to re-evaluate our approach since this leads to a contradiction.

### Step 2: Rearranging the equation
Let's rearrange the original equation:
\[ 2x - 3 = 2x \]
This is incorrect. Instead, we should have:
\[ 2x - 3 = 2x \implies -3 = 0 \text{ (which is not possible)} \]
So, let's go back to the original equation:
\[ 2x - 3 = 2x \]
This leads us to realize that we need to isolate \( x \).

### Step 3: Rewrite the equation correctly
Revisiting the equation:
\[ 2x - 3 = 2x \]
This is incorrect. Let's try again:
\[ 2x - 3 = 2x \implies -3 = 0 \text{ (which is not possible)} \]
We should have:
\[ 2x - 3 = 2x \implies -3 = 0 \text{ (which is not possible)} \]
This indicates we need to isolate \( x \).

### Step 4: Squaring both sides
We can square both sides of the equation:
\[ \left(\frac{2x - 3}{x}\right)^2 = 2^2 \]
This gives us:
\[ \frac{(2x - 3)^2}{x^2} = 4 \]
Multiplying both sides by \( x^2 \):
\[ (2x - 3)^2 = 4x^2 \]

### Step 5: Expand and simplify
Expanding the left side:
\[ 4x^2 - 12x + 9 = 4x^2 \]
Subtract \( 4x^2 \) from both sides:
\[ -12x + 9 = 0 \]
Rearranging gives:
\[ 12x = 9 \implies x = \frac{3}{4} \]

### Step 6: Substitute \( x \) into the expression
Now we need to find \( 16x^4 + \frac{81}{x^4} \):
First, calculate \( x^4 \):
\[ x^4 = \left(\frac{3}{4}\right)^4 = \frac{81}{256} \]
Now substitute into the expression:
\[ 16x^4 = 16 \times \frac{81}{256} = \frac{1296}{256} \]
And for \( \frac{81}{x^4} \):
\[ \frac{81}{x^4} = \frac{81}{\frac{81}{256}} = 256 \]

### Step 7: Combine the two results
Now we combine:
\[ 16x^4 + \frac{81}{x^4} = \frac{1296}{256} + 256 \]
Convert \( 256 \) to a fraction:
\[ 256 = \frac{65536}{256} \]
Now add:
\[ \frac{1296 + 65536}{256} = \frac{66832}{256} \]

### Step 8: Simplify the fraction
Now simplify:
\[ 66832 \div 256 = 261 \]

Thus, the final answer is:
\[ \boxed{261} \]