The expression ` x^(4) + 2 x ^(2) + k` will be a perfect square when the value of k is

Options

A. A)2
B. B)1
C. C)`-1`
D. D)`-2`

Answer

To determine the value of \( k \) for which the expression \( x^4 + 2x^2 + k \) is a perfect square, we can follow these steps:

### Step 1: Identify the structure of the expression
The expression \( x^4 + 2x^2 + k \) resembles a quadratic in terms of \( x^2 \). We can rewrite it as:
\[ (x^2)^2 + 2(x^2) + k \]
This suggests that we can treat \( x^2 \) as a single variable, say \( y \). Thus, we rewrite the expression as:
\[ y^2 + 2y + k \]

### Step 2: Apply the condition for a perfect square
For a quadratic expression of the form \( ay^2 + by + c \) to be a perfect square, the discriminant must be zero. The discriminant \( D \) is given by:
\[ D = b^2 - 4ac \]
In our case, \( a = 1 \), \( b = 2 \), and \( c = k \). Therefore, the discriminant becomes:
\[ D = 2^2 - 4 \cdot 1 \cdot k \]
This simplifies to:
\[ D = 4 - 4k \]

### Step 3: Set the discriminant to zero
For the expression to be a perfect square, we need the discriminant to equal zero:
\[ 4 - 4k = 0 \]

### Step 4: Solve for \( k \)
Now we solve the equation:
\[ 4 = 4k \]
Dividing both sides by 4 gives:
\[ 1 = k \]

### Conclusion
Thus, the value of \( k \) that makes the expression \( x^4 + 2x^2 + k \) a perfect square is:
\[ \boxed{1} \]