The number of real solutions of `x-1/(x^2-4)=2-1/(x^2-4)` is :

Options

A. 0
B. 1
C. 2
D. infinite

Answer

To solve the equation \( x - \frac{1}{x^2 - 4} = 2 - \frac{1}{x^2 - 4} \), we can follow these steps:

### Step 1: Rewrite the equation
Start by rewriting the equation clearly:
\[ x - \frac{1}{x^2 - 4} = 2 - \frac{1}{x^2 - 4} \]

### Step 2: Simplify the equation
Since both sides have the term \( -\frac{1}{x^2 - 4} \), we can add this term to both sides:
\[ x = 2 \]

### Step 3: Identify restrictions
Now, we need to consider the restrictions on \( x \). The term \( x^2 - 4 \) cannot be zero because it is in the denominator. Thus, we set:
\[ x^2 - 4 \neq 0 \]
This simplifies to:
\[ x^2 \neq 4 \]
which gives us:
\[ x \neq 2 \quad \text{and} \quad x \neq -2 \]

### Step 4: Analyze the solution
From Step 2, we found that \( x = 2 \) is a solution. However, from Step 3, we see that \( x = 2 \) is not allowed because it makes the denominator zero.

### Conclusion
Since the only solution we found is not valid due to the restriction, we conclude that there are no real solutions to the equation.

### Final Answer
The number of real solutions is **0**.

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