If ` x + (1)/( 9 x) = 4` then the value ` 9 x^(2) + (1)/( 9 x^(2) ` is

Options

A. 140
B. 142
C. 144
D. 146

Answer

To solve the equation \( x + \frac{1}{9x} = 4 \) and find the value of \( 9x^2 + \frac{1}{9x^2} \), we can follow these steps:

### Step 1: Isolate \( x \)
We start with the equation:
\[ x + \frac{1}{9x} = 4 \]
To eliminate the fraction, we can multiply the entire equation by \( 9x \):
\[ 9x^2 + 1 = 36x \]

### Step 2: Rearrange the equation
Now, we rearrange the equation to form a standard quadratic equation:
\[ 9x^2 - 36x + 1 = 0 \]

### Step 3: Solve the quadratic equation
We can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 9, b = -36, c = 1 \):
\[ b^2 - 4ac = (-36)^2 - 4 \cdot 9 \cdot 1 = 1296 - 36 = 1260 \]
Now, substituting into the quadratic formula:
\[ x = \frac{36 \pm \sqrt{1260}}{18} \]

### Step 4: Simplify \( \sqrt{1260} \)
We can simplify \( \sqrt{1260} \):
\[ \sqrt{1260} = \sqrt{36 \times 35} = 6\sqrt{35} \]
Thus, we have:
\[ x = \frac{36 \pm 6\sqrt{35}}{18} = 2 \pm \frac{\sqrt{35}}{3} \]

### Step 5: Find \( 9x^2 + \frac{1}{9x^2} \)
We can use the identity:
\[ 9x^2 + \frac{1}{9x^2} = \left( x + \frac{1}{9x} \right)^2 - 2 \]
From the original equation, we know \( x + \frac{1}{9x} = 4 \). Therefore:
\[ 9x^2 + \frac{1}{9x^2} = 4^2 - 2 = 16 - 2 = 14 \]

### Final Answer
Thus, the value of \( 9x^2 + \frac{1}{9x^2} \) is:
\[ \boxed{14} \]