If `x + (1)/(x) = 4sqrt(3)`, then find the value of `x^(2) + (1)/(x^(2))` ?

Options

A. A)46
B. B)44
C. C)56
D. D)52

Answer

To solve the equation \( x + \frac{1}{x} = 4\sqrt{3} \) 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:
\[ x + \frac{1}{x} = 4\sqrt{3} \]
Now, we square both sides:
\[ \left( x + \frac{1}{x} \right)^2 = (4\sqrt{3})^2 \]

### Step 2: Expand the left side
Using the identity \( (a + b)^2 = a^2 + 2ab + b^2 \), we expand the left side:
\[ x^2 + 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^2} = 48 \]
Since \( x \cdot \frac{1}{x} = 1 \), we can simplify this to:
\[ x^2 + 2 + \frac{1}{x^2} = 48 \]

### Step 3: Rearrange the equation
Now, we can rearrange the equation to isolate \( x^2 + \frac{1}{x^2} \):
\[ x^2 + \frac{1}{x^2} = 48 - 2 \]

### Step 4: Simplify the result
Calculating the right side gives us:
\[ x^2 + \frac{1}{x^2} = 46 \]

Thus, the value of \( x^2 + \frac{1}{x^2} \) is \( \boxed{46} \).

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