Consider the equation $x^4 - (k-1)x^2 + (2-k) = 0$. The complete set of possible values of real $k$ for which the equation has four distinct real roots is:
Step-by-Step Solution
Key Concept: Graphical analysis of $k$ as a function of $t$ determines the number and nature of roots in both $t$ and the original variable $x$.
Substituting $x^2 = t$ transforms the original equation into $t^2 - (k-1)t + 2-k = 0$. From the graph of $k = \frac{t^2+t+2}{t+1}$ for $t \geq 0$, determine which values of $k$ yield 2 real and distinct roots: $k \in [2\sqrt{2}-1] \cup (2, \infty)$ gives 2 distinct real roots in $x$; $k \in \{2, 3\}$ gives 3 distinct real roots; $k \in [2\sqrt{2}-1.2)$ gives 4 distinct real roots.
Correct Answer: 2