If the equation x^2 - (k-1)|x| + (k+2) = 0 has exactly four distinct real roots, then the range of values for the parameter k is :

Step-by-step solution

Let t = |x| , where t 0 . The given equation becomes a quadratic in t : t^2 - (k-1)t + (k+2) = 0 For the original equation in x to have exactly four distinct real roots, the quadratic equation in t must have exactly two distinct positive real roots. This requires three conditions to be satisfied simultaneously: 1) Discriminant D > 0 : (k-1)^2 - 4(1)(k+2) > 0 k^2 - 2k + 1 - 4k - 8 > 0 k^2 - 6k - 7 > 0 (k-7)(k+1) > 0 k (- , -1) (7, ) 2) Sum of roots > 0 : -(-(k-1)) 1 > 0 k - 1 > 0 k > 1 3) Product of roots > 0 : k+2