The equation `(x^(2))/(|x-2|)=|(2x)/(x-2)|+|x|` has solutions whose number is

Options

A. `1`
B. `2`
C. `0`
D. `oo`

Answer

To solve the equation \(\frac{x^2}{|x-2|} = \left|\frac{2x}{x-2}\right| + |x|\), we will analyze it by considering different cases based on the value of \(x\).

### Step 1: Consider the case when \(x > 2\)

In this case, \(|x-2| = x-2\) and \(\left|\frac{2x}{x-2}\right| = \frac{2x}{x-2}\) since \(x-2\) is positive. Thus, the equation becomes:

\[ \frac{x^2}{x-2} = \frac{2x}{x-2} + x \]

### Step 2: Simplify the equation

We can multiply through by \(x-2\) (valid since \(x > 2\)):

\[ x^2 = 2x + x(x-2) \]

Expanding the right-hand side:

\[ x^2 = 2x + x^2 - 2x \]

This simplifies to:

\[ x^2 = x^2 \]

### Step 3: Analyze the result

The equation \(x^2 = x^2\) is always true for any \(x > 2\). Therefore, every \(x\) greater than 2 is a solution.

### Step 4: Conclusion about the number of solutions

Since there are infinitely many values of \(x\) that satisfy \(x > 2\) (e.g., \(x = 3, 4, 5, \ldots\)), we conclude that the number of solutions is infinite.

Thus, the answer is:

**The number of solutions is infinite.**