Consider equation `x^(4)-6x^(3)+8x^(2)+4ax-4a^(2)=0,ainR`. Then match the following lists:

Options

A. `{:(,a,b,c,d),((1),q,s,s,r):}`
B. `{:(,a,b,c,d),((2),r,s,q,p):}`
C. `{:(,a,b,c,d),((3),q,s,r,p):}`
D. `{:(,a,b,c,d),((4),q,r,p,p):}`

Answer

4

Explanation

`(x^(2)-2x-2a)(x^(2)-4x+2a)=0`
`impliesx^(2)-2x-2a=0" "....(1)` or
`x^(2)-4x+2a=0" "....(2)`
Discrtiminant of eq. (1) is: `D_(1)=4+8a`
Discriminant of eq. (2) is: `D_(2)=16-8a`
(a) If equation has exactly two distinct roots then
(1) `D_(1)gt0implies4+8agt0impliesagt-1//2` and
`D_(2)implies16-8agt0impliesalt2`
`:.ain(-1//2,2)`
(b) If equation has exactly two distinct roots then (i) `D_(1)gt0andD_(2)lt0`
`impliesain(2,oo)`
(ii) `D_(1)lt0andD_(2)gt0`
`impliesain(-oo,-1//2)`
(c) If equation has no real roots then
`D_(1)lt0andD_(2)lt0`
`impliesainphi`
(d) Clearly, rquation cannot have four roots positive.