If $x = \alpha$ is the common root of both the equations, then find $P(1)$ where $P(x) = x^2 - 2x + 5$.
Step-by-Step Solution
Key Concept: A common root satisfies both equations simultaneously; eliminate the higher powers using the first equation to find the quadratic.
If $\alpha$ is a common root of $x^4 + 6x^2 + 25 = 0$ and another equation, then $\alpha^4 + 6\alpha^2 + 25 = 0 \Rightarrow \alpha^4 = -6\alpha^2 - 25$. Substituting into the second equation and simplifying yields $\alpha^2 - 2\alpha + 5 = 0$, so $P(\alpha) = 0$. Therefore, $P(x) = x^2 - 2x + 5$ and $P(1) = 1 - 2 + 5 = 4$.
Correct Answer: 4