If `sqrt(x)+(1)/(sqrt(x)) =sqrt(6)`, then `x^(2)+(1)/(x^(2))` is equal to :

Options

A. 16
B. 62
C. 36
D. 14

Answer

To solve the equation \( \sqrt{x} + \frac{1}{\sqrt{x}} = \sqrt{6} \) and find the value of \( x^2 + \frac{1}{x^2} \), we can follow these steps:

### Step 1: Square both sides of the equation
We start with the equation:
\[ \sqrt{x} + \frac{1}{\sqrt{x}} = \sqrt{6} \]
Now, we will square both sides:
\[ \left(\sqrt{x} + \frac{1}{\sqrt{x}}\right)^2 = (\sqrt{6})^2 \]

### Step 2: Expand the left side
Expanding the left side gives:
\[ \left(\sqrt{x}\right)^2 + 2 \cdot \sqrt{x} \cdot \frac{1}{\sqrt{x}} + \left(\frac{1}{\sqrt{x}}\right)^2 = 6 \]
This simplifies to:
\[ x + 2 + \frac{1}{x} = 6 \]

### Step 3: Rearrange the equation
Now, we can rearrange the equation:
\[ x + \frac{1}{x} + 2 = 6 \]
Subtracting 2 from both sides gives:
\[ x + \frac{1}{x} = 4 \]

### Step 4: Square again to find \( x^2 + \frac{1}{x^2} \)
Next, we square both sides again:
\[ \left(x + \frac{1}{x}\right)^2 = 4^2 \]
Expanding the left side:
\[ x^2 + 2 + \frac{1}{x^2} = 16 \]

### Step 5: Isolate \( x^2 + \frac{1}{x^2} \)
Now, we isolate \( x^2 + \frac{1}{x^2} \):
\[ x^2 + \frac{1}{x^2} + 2 = 16 \]
Subtracting 2 from both sides gives:
\[ x^2 + \frac{1}{x^2} = 16 - 2 \]
Thus,
\[ x^2 + \frac{1}{x^2} = 14 \]

### Final Answer
The value of \( x^2 + \frac{1}{x^2} \) is \( 14 \).