The equation `x-(3)/(x-2) =2-(3)/(x-2)` has

Options

A. No root
B. One root
C. Two roots
D. infinite roots

Answer

To solve the equation \( x - \frac{3}{x - 2} = 2 - \frac{3}{x - 2} \), we will follow these steps:

### Step 1: Set up the equation
We start with the equation:
\[ x - \frac{3}{x - 2} = 2 - \frac{3}{x - 2} \]

### Step 2: Eliminate the fractions
To eliminate the fractions, we can multiply both sides of the equation by \( x - 2 \) (noting that \( x \neq 2 \) to avoid division by zero):
\[ (x - 2) \left( x - \frac{3}{x - 2} \right) = (x - 2) \left( 2 - \frac{3}{x - 2} \right) \]

This simplifies to:
\[ (x - 2)x - 3 = (x - 2)2 - 3 \]

### Step 3: Expand both sides
Expanding both sides gives:
\[ x^2 - 2x - 3 = 2x - 4 - 3 \]
\[ x^2 - 2x - 3 = 2x - 7 \]

### Step 4: Rearrange the equation
Now, we can rearrange the equation to bring all terms to one side:
\[ x^2 - 2x - 3 - 2x + 7 = 0 \]
\[ x^2 - 4x + 4 = 0 \]

### Step 5: Factor the quadratic equation
The quadratic can be factored as:
\[ (x - 2)^2 = 0 \]

### Step 6: Solve for x
Setting the factored equation to zero gives:
\[ x - 2 = 0 \implies x = 2 \]

### Step 7: Check for validity
However, we must check if \( x = 2 \) is a valid solution. Since we initially stated that \( x \neq 2 \) (as it would make the original equation undefined), we conclude that there are no valid roots.

### Final Conclusion
Thus, the equation has **no roots**.

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