if `sqrtx + (1)/(sqrt(x)) = 4`, find the value of `x^(2) + (1)/(x^(2))`.

Options

A. 196
B. 258
C. 254
D. 194

Answer

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

### Step 1: Let \( y = \sqrt{x} \)
We start by substituting \( y = \sqrt{x} \). This gives us:
\[ y + \frac{1}{y} = 4 \]

### Step 2: Square both sides
Next, we square both sides of the equation:
\[ \left(y + \frac{1}{y}\right)^2 = 4^2 \]
This simplifies to:
\[ y^2 + 2 \cdot y \cdot \frac{1}{y} + \frac{1}{y^2} = 16 \]
\[ y^2 + 2 + \frac{1}{y^2} = 16 \]

### Step 3: Rearrange the equation
Now, we can rearrange the equation to isolate \( y^2 + \frac{1}{y^2} \):
\[ y^2 + \frac{1}{y^2} = 16 - 2 \]
\[ y^2 + \frac{1}{y^2} = 14 \]

### Step 4: Relate back to \( x \)
Since \( y = \sqrt{x} \), we have:
\[ \sqrt{x}^2 + \frac{1}{\sqrt{x}^2} = x + \frac{1}{x} \]
Thus,
\[ x + \frac{1}{x} = 14 \]

### Step 5: Square again to find \( x^2 + \frac{1}{x^2} \)
Now, we square \( x + \frac{1}{x} \):
\[ \left(x + \frac{1}{x}\right)^2 = 14^2 \]
This gives:
\[ x^2 + 2 + \frac{1}{x^2} = 196 \]
\[ x^2 + \frac{1}{x^2} = 196 - 2 \]
\[ x^2 + \frac{1}{x^2} = 194 \]

### Final Answer
Thus, the value of \( x^2 + \frac{1}{x^2} \) is:
\[ \boxed{194} \]