If a = 0 then the equation `(x-a-1)/(x-a)= (a +1)-(1)/(x-a)` has

Options

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

Answer

To solve the given equation \(\frac{x-a-1}{x-a} = (a + 1) - \frac{1}{x-a}\) under the condition that \(a = 0\), we will follow these steps:

### Step 1: Substitute \(a = 0\) into the equation
The given equation is:
\[ \frac{x-a-1}{x-a} = (a + 1) - \frac{1}{x-a} \]
Substituting \(a = 0\):
\[ \frac{x - 0 - 1}{x - 0} = (0 + 1) - \frac{1}{x - 0} \]
This simplifies to:
\[ \frac{x - 1}{x} = 1 - \frac{1}{x} \]

### Step 2: Simplify both sides
Now we simplify both sides:
Left-hand side:
\[ \frac{x - 1}{x} = 1 - \frac{1}{x} \]
Right-hand side:
\[ 1 - \frac{1}{x} = 1 - \frac{1}{x} \]
Both sides are equal, so we have:
\[ \frac{x - 1}{x} = 1 - \frac{1}{x} \]

### Step 3: Analyze the equation
The equation holds true for all \(x\) except where the denominator is zero. The denominator \(x - 0\) becomes zero when \(x = 0\). Thus, the equation is valid for all real numbers except \(x = 0\).

### Step 4: Conclusion about the roots
Since the equation holds for all \(x\) except \(x = 0\), it implies that there are infinitely many solutions (or roots) for the equation in the set of real numbers excluding zero.

### Final Answer
Therefore, the equation has **many roots**.

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