Answer: D
The expression \(x^4 - 2x^2 + k\) will be a perfect square when the equation \(x^4 - 2x^2 + k = 0\) has equal roots, i.e., the discriminant is zero. \[ \therefore (-2)^2 - 4 \times k \times 1 = 0 \Rightarrow 4 - 4k = 0 \Rightarrow -4k = -4 \Rightarrow k = 1 \] --- **Alternatively:** \[ x^4 - 2x^2 + k = (x^2)^2 - 2 \cdot x^2 \cdot 1 + 1^2 - 1^2 + k = (x^2 - 1)^2 - 1 + k \] The expression will be a perfect square when: \[ -1 + k = 0 \Rightarrow k = 1 \]
The expression \(x^4 - 2x^2 + k\) will be a perfect square when the equation \(x^4 - 2x^2 + k = 0\) has equal roots, i.e., the discriminant is zero. \[ \therefore (-2)^2 - 4 \times k \times 1 = 0 \Rightarrow 4 - 4k = 0 \Rightarrow -4k = -4 \Rightarrow k = 1 \] --- **Alternatively:** \[ x^4 - 2x^2 + k = (x^2)^2 - 2 \cdot x^2 \cdot 1 + 1^2 - 1^2 + k = (x^2 - 1)^2 - 1 + k \] The expression will be a perfect square when: \[ -1 + k = 0 \Rightarrow k = 1 \]