If `x - (1)/( x) = sqrt21,` then the value of `(x ^(2) + (1)/( x ^(2))) (x + (1)/(x))` is

Options

A. 113
B. 115
C. 114
D. None of these

Answer

To solve the equation \( x - \frac{1}{x} = \sqrt{21} \) and find the value of \( (x^2 + \frac{1}{x^2})(x + \frac{1}{x}) \), we can follow these steps:

### Step 1: Square both sides of the equation
Starting with the equation:
\[ x - \frac{1}{x} = \sqrt{21} \]
We square both sides:
\[ \left(x - \frac{1}{x}\right)^2 = (\sqrt{21})^2 \]
This gives us:
\[ x^2 - 2\cdot x \cdot \frac{1}{x} + \frac{1}{x^2} = 21 \]
Simplifying, we have:
\[ x^2 - 2 + \frac{1}{x^2} = 21 \]

### Step 2: Rearrange to find \( x^2 + \frac{1}{x^2} \)
Now, we rearrange the equation:
\[ x^2 + \frac{1}{x^2} = 21 + 2 \]
Thus, we find:
\[ x^2 + \frac{1}{x^2} = 23 \]

### Step 3: Find \( x + \frac{1}{x} \)
Next, we need to find \( x + \frac{1}{x} \). We can use the identity:
\[ \left(x + \frac{1}{x}\right)^2 = x^2 + 2 + \frac{1}{x^2} \]
Let \( y = x + \frac{1}{x} \). Then:
\[ y^2 = x^2 + 2 + \frac{1}{x^2} \]
Substituting \( x^2 + \frac{1}{x^2} \):
\[ y^2 = 23 + 2 \]
This simplifies to:
\[ y^2 = 25 \]
Taking the square root gives:
\[ y = 5 \quad \text{(since \( x + \frac{1}{x} \) is positive)} \]

### Step 4: Calculate the final expression
Now we need to calculate:
\[ (x^2 + \frac{1}{x^2})(x + \frac{1}{x}) = 23 \cdot 5 \]
Calculating this gives:
\[ 23 \cdot 5 = 115 \]

### Final Answer
Thus, the value of \( (x^2 + \frac{1}{x^2})(x + \frac{1}{x}) \) is \( \boxed{115} \).

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