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

Answer: 5

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 $-1,0,1,2,3$