Find the value of k, if
`(x^(2))/((x^(2)+2)^(2))=(1)/(x^(2)+2)+(k)/((x^(2)+2)^(2))`

Answer

To solve the equation

\[ \frac{x^2}{(x^2 + 2)^2} = \frac{1}{x^2 + 2} + \frac{k}{(x^2 + 2)^2}, \]

we will find the value of \( k \).

### Step 1: Multiply both sides by \((x^2 + 2)^2\)

We start by eliminating the denominators. We multiply both sides of the equation by \((x^2 + 2)^2\):

\[ x^2 = (x^2 + 2) + k. \]

### Step 2: Simplify the equation

Now, we simplify the right-hand side:

\[ x^2 = x^2 + 2 + k. \]

### Step 3: Rearrange the equation

Next, we rearrange the equation to isolate \( k \):

\[ x^2 - (x^2 + 2 + k) = 0. \]

This simplifies to:

\[ 0 = 2 + k. \]

### Step 4: Solve for \( k \)

Now, we solve for \( k \):

\[ k = -2. \]

### Final Answer

Thus, the value of \( k \) is

\[ \boxed{-2}. \]

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