If `x^(2)+1/(x^(2))=83`, the value of `x-1/(x)` is

Options

A. 9
B. `sqrt(85)`
C. 81
D. 85

Answer

To solve the equation \( x^2 + \frac{1}{x^2} = 83 \) and find the value of \( x - \frac{1}{x} \), we can follow these steps:

### Step 1: Use the identity
We will use the identity:
\[ (a - b)^2 = a^2 + b^2 - 2ab \]
In our case, let \( a = x \) and \( b = \frac{1}{x} \). Then, we can express \( (x - \frac{1}{x})^2 \) as:
\[ (x - \frac{1}{x})^2 = x^2 + \frac{1}{x^2} - 2 \]

### Step 2: Substitute the known value
We know from the problem that:
\[ x^2 + \frac{1}{x^2} = 83 \]
Substituting this into our identity gives:
\[ (x - \frac{1}{x})^2 = 83 - 2 \]

### Step 3: Simplify the equation
Now, simplify the right side:
\[ (x - \frac{1}{x})^2 = 81 \]

### Step 4: Take the square root
To find \( x - \frac{1}{x} \), we take the square root of both sides:
\[ x - \frac{1}{x} = \sqrt{81} \]

### Step 5: Calculate the square root
Calculating the square root:
\[ x - \frac{1}{x} = 9 \]

Thus, the value of \( x - \frac{1}{x} \) is \( 9 \).

### Final Answer
The value of \( x - \frac{1}{x} \) is \( 9 \).

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