If `sqrt(x)-(1)/(sqrt(x))=sqrt(5)`, then `(x^(2)+(1)/(x^(2)))` is equal to :

Options

A. 45
B. 49
C. 47
D. 51

Answer

To solve the equation \( \sqrt{x} - \frac{1}{\sqrt{x}} = \sqrt{5} \) 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:
\[ \sqrt{x} - \frac{1}{\sqrt{x}} = \sqrt{5} \]
Now, we square both sides:
\[ \left(\sqrt{x} - \frac{1}{\sqrt{x}}\right)^2 = (\sqrt{5})^2 \]

### Step 2: Expand the left side
Using the formula \( (a - b)^2 = a^2 - 2ab + b^2 \), we can expand the left side:
\[ \left(\sqrt{x}\right)^2 - 2\left(\sqrt{x}\right)\left(\frac{1}{\sqrt{x}}\right) + \left(\frac{1}{\sqrt{x}}\right)^2 = 5 \]
This simplifies to:
\[ x - 2 + \frac{1}{x} = 5 \]

### Step 3: Rearrange the equation
Now, we can rearrange the equation to isolate \( x + \frac{1}{x} \):
\[ x + \frac{1}{x} = 5 + 2 \]
\[ x + \frac{1}{x} = 7 \]

### Step 4: Square the result
Next, we square \( x + \frac{1}{x} \) to find \( x^2 + \frac{1}{x^2} \):
\[ \left(x + \frac{1}{x}\right)^2 = x^2 + 2 + \frac{1}{x^2} \]
Substituting \( 7 \) into the equation:
\[ 7^2 = x^2 + 2 + \frac{1}{x^2} \]
\[ 49 = x^2 + 2 + \frac{1}{x^2} \]

### Step 5: Solve for \( x^2 + \frac{1}{x^2} \)
Now, we can isolate \( x^2 + \frac{1}{x^2} \):
\[ x^2 + \frac{1}{x^2} = 49 - 2 \]
\[ x^2 + \frac{1}{x^2} = 47 \]

### Final Answer
Thus, the value of \( x^2 + \frac{1}{x^2} \) is:
\[ \boxed{47} \]