The equation `x-2(x-1)^(-1)=1-2(x-1)^(-1)` has

Options

A. no roots
B. One root is real and the other is complex
C. two equal roots
D. infinite roots

Explanation

To solve the equation \( x - 2(x-1)^{-1} = 1 - 2(x-1)^{-1} \), we will follow these steps:

### Step 1: Simplify the equation

Start with the given equation:

\[ x - 2(x-1)^{-1} = 1 - 2(x-1)^{-1} \]

### Step 2: Move all terms involving \((x-1)^{-1}\) to one side

We can add \(2(x-1)^{-1}\) to both sides:

\[ x = 1 \]

### Step 3: Substitute back to check for restrictions

Now, we need to check if substituting \(x = 1\) into the original equation causes any issues, particularly with the term \((x-1)^{-1}\):

\[ (x-1)^{-1} \text{ becomes } (1-1)^{-1} = 0^{-1} \]

This is undefined.

### Step 4: Conclusion

Since substituting \(x = 1\) leads to an undefined expression, the equation has no valid solutions.

### Final Answer

The equation has no roots.

---