Find the roots of `x+(1)/(x)=2`

Answer

To find the roots of the equation \( x + \frac{1}{x} = 2 \), we can follow these steps:

### Step 1: Eliminate the fraction
Multiply both sides of the equation by \( x \) (assuming \( x \neq 0 \)):
\[ x^2 + 1 = 2x \]

### Step 2: Rearrange the equation
Rearranging the equation gives:
\[ x^2 - 2x + 1 = 0 \]

### Step 3: Factor the quadratic equation
The equation \( x^2 - 2x + 1 = 0 \) can be factored as:
\[ (x - 1)(x - 1) = 0 \]
or simply:
\[ (x - 1)^2 = 0 \]

### Step 4: Solve for \( x \)
Setting the factored form equal to zero gives:
\[ x - 1 = 0 \]
Thus, we find:
\[ x = 1 \]

### Step 5: Identify the roots
Since the factor \( (x - 1) \) is repeated, the root is:
\[ x = 1 \]
This means that the equation has a double root at \( x = 1 \).

### Final Answer
The roots of the equation \( x + \frac{1}{x} = 2 \) are \( x = 1 \) (with multiplicity 2).

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