If `(3x-(2)/(x))=5`, then the value of `(9x^(2)-(4)/(x^(2)))` is :

Options

A. 25
B. 35
C. 30
D. none of these

Answer

To solve the equation \( 3x - \frac{2}{x} = 5 \) and find the value of \( 9x^2 - \frac{4}{x^2} \), we can follow these steps:

### Step 1: Rearranging the Equation
Start with the given equation:
\[ 3x - \frac{2}{x} = 5 \]
Rearranging gives:
\[ 3x = 5 + \frac{2}{x} \]

### Step 2: Multiply Both Sides by \( x \)
To eliminate the fraction, multiply both sides by \( x \):
\[ 3x^2 = 5x + 2 \]

### Step 3: Rearranging into Standard Form
Rearranging the equation leads to:
\[ 3x^2 - 5x - 2 = 0 \]

### Step 4: Solving the Quadratic Equation
Now, we can apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 3, b = -5, c = -2 \):
\[ b^2 - 4ac = (-5)^2 - 4 \cdot 3 \cdot (-2) = 25 + 24 = 49 \]
Thus, the roots are:
\[ x = \frac{5 \pm \sqrt{49}}{2 \cdot 3} = \frac{5 \pm 7}{6} \]
Calculating the two possible values for \( x \):
1. \( x = \frac{12}{6} = 2 \)
2. \( x = \frac{-2}{6} = -\frac{1}{3} \)

### Step 5: Finding \( 9x^2 - \frac{4}{x^2} \)
Now, we need to find \( 9x^2 - \frac{4}{x^2} \). First, calculate \( x^2 \):
- For \( x = 2 \):
\[ x^2 = 2^2 = 4 \]
Then:
\[ 9x^2 = 9 \cdot 4 = 36 \]
And:
\[ \frac{4}{x^2} = \frac{4}{4} = 1 \]
Thus:
\[ 9x^2 - \frac{4}{x^2} = 36 - 1 = 35 \]

- For \( x = -\frac{1}{3} \):
\[ x^2 = \left(-\frac{1}{3}\right)^2 = \frac{1}{9} \]
Then:
\[ 9x^2 = 9 \cdot \frac{1}{9} = 1 \]
And:
\[ \frac{4}{x^2} = \frac{4}{\frac{1}{9}} = 36 \]
Thus:
\[ 9x^2 - \frac{4}{x^2} = 1 - 36 = -35 \]

### Final Answer
The values of \( 9x^2 - \frac{4}{x^2} \) are \( 35 \) for \( x = 2 \) and \( -35 \) for \( x = -\frac{1}{3} \).