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

Options

A. 62
B. 40
C. 54
D. 66

Answer

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

### Step 1: Let \( y = \sqrt{x} \)
This substitution simplifies our equation. Thus, we rewrite the equation as:
\[ y - \frac{1}{y} = \sqrt{6} \]

### Step 2: Multiply both sides by \( y \)
To eliminate the fraction, we multiply through by \( y \):
\[ y^2 - 1 = y\sqrt{6} \]

### Step 3: Rearrange the equation
Rearranging gives us a standard quadratic form:
\[ y^2 - y\sqrt{6} - 1 = 0 \]

### Step 4: Use the quadratic formula
We can solve for \( y \) using the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = -\sqrt{6}, c = -1 \):
\[ y = \frac{\sqrt{6} \pm \sqrt{(\sqrt{6})^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} \]
\[ y = \frac{\sqrt{6} \pm \sqrt{6 + 4}}{2} \]
\[ y = \frac{\sqrt{6} \pm \sqrt{10}}{2} \]

### Step 5: Find \( y^2 \)
Next, we need to calculate \( y^2 \):
\[ y^2 = \left( \frac{\sqrt{6} + \sqrt{10}}{2} \right)^2 = \frac{6 + 10 + 2\sqrt{60}}{4} = \frac{16 + 2\sqrt{60}}{4} = 4 + \frac{\sqrt{60}}{2} \]

### Step 6: Calculate \( x^2 + \frac{1}{x^2} \)
Now, we need to find \( x^2 + \frac{1}{x^2} \):
\[ x^2 + \frac{1}{x^2} = y^2 + \frac{1}{y^2} \]
Using the identity \( \frac{1}{y^2} = \frac{1}{\left( \frac{\sqrt{6} + \sqrt{10}}{2} \right)^2} \):
\[ \frac{1}{y^2} = \frac{4}{16 + 2\sqrt{60}} = \frac{4}{4(4 + \frac{\sqrt{60}}{2})} = \frac{1}{4 + \frac{\sqrt{60}}{2}} \]

### Step 7: Combine the results
Using the identity \( y^2 + \frac{1}{y^2} = \left( y - \frac{1}{y} \right)^2 + 2 \):
\[ x^2 + \frac{1}{x^2} = \left( \sqrt{6} \right)^2 + 2 = 6 + 2 = 8 \]

### Final Answer
Thus, \( x^2 + \frac{1}{x^2} = 8 \).

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