Question

Find the number of integral values of $k$ for which the equation $x^{2}-4|x|+3-|k-1|=0$ has four distinct real roots.

Answer 1

Given $| x |^{2}-4| x |+3-| k -1|=0$...(1)
As above equation (1) have four distinct real roots so both of equation (1) must be positive and distinct Now,
(i) $D >0 \Rightarrow 16-12+4| k -1|>0$
$\Rightarrow 1+| k -1|>0$, which is true $\forall k \in R$.
(ii) Sum of roots $>0 \Rightarrow 4>0$, which is true $\forall k \in R$.
(iii) Product of roots $>0 \Rightarrow 3-| k -1|>0$
$\Rightarrow | k -1|<3 \Rightarrow -2< k <4$
must be satisfied simultaneously.
$\therefore 1 \cap 2 \cap 3 \Rightarrow k \in(-2,4)$
Clearly, possible integral values of $k$ are $-1,0,1,2,3$