If `alpha_(1),alpha_(2),alpha_(3), alpha_(4)` are the roos of `x^(4)+2x^(3)+bx^(2)+cx+d=0` such that `alpha_(1)-alpha_(3)=alpha_(4)-alpha_(2)`, then `b-c` is equal to__________

Answer

1

Explanation

`(x^(4)+2x^(3)+bx^(2)+cx+d)=(x-alpha_(1))(x-alpha_(2))(x-alpha_(3))(x-alpha_(4))=(x^(2)+px+q)(x^(2)+px+r)`
`=x^(4)+2px^(3)+(q+r+p^(2))x^(2)+(pr+pq)x+qr`
`(2p)^(3)-4(2p)(q+r+p^(2))+8(pq+pr)=0`
`2^(3)-4xx2xxb+8c=0`
`impliesb-c=1`