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

Options

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

Answer

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

### Step 1: Identify the form of a perfect square
A perfect square trinomial can be expressed in the form \( (a - b)^2 = a^2 - 2ab + b^2 \).

### Step 2: Rewrite the expression
We can rewrite the given expression \( x^4 - 2x^2 + k \) in a way that resembles the perfect square format. Notice that \( x^4 \) can be written as \( (x^2)^2 \) and \( -2x^2 \) can be related to the term \( -2ab \).

### Step 3: Match the terms
Let \( a = x^2 \) and \( b = \sqrt{k} \). Then we have:
- \( a^2 = (x^2)^2 = x^4 \)
- \( -2ab = -2(x^2)(\sqrt{k}) = -2x^2\sqrt{k} \)

### Step 4: Set up the equation
For the expression to be a perfect square, we need:
\[ -2x^2\sqrt{k} = -2x^2 \]
This implies:
\[ \sqrt{k} = 1 \]

### Step 5: Solve for \( k \)
Squaring both sides gives:
\[ k = 1^2 = 1 \]

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

---