Suppose the quadratic polynomial `p(x) = ax^2 + bx + c` has positive coefficient `a, b, c` such that `b- a=c-b`. If `p(x) = 0` has integer roots `alpha and beta` then what could be the possible value of `alpha+beta+alpha beta` if `0 leq alpha+beta+alpha beta leq 8`

Options

A. 3
B. 5
C. 7
D. 14

Answer

C

Explanation

`P(x)=ax^(2)+bx+c=a(x-alpha)9x-beta)`
and `alpha+beta+alphabeta+1-1=(alpha=1)(beta=1)-1`
`((a-b+c))/a-1`
`Rightarrowalpha+beta+alphabeta=b/a-1=lambda_(1)-1`
i.e. ,` b/a` is interger= `lambda_(1)`
if b = `alambda_(1)`
then `c=a(2lambda_(1)-1)` (because a,b,c are in A.P)
`P(x)=ax^(2)=alambda_(1)x+a(2lambda_(1)-1)`
`a[x^(2)+lambda_(1)x+(2lambda_(1)-1)]`
`D=lambda_(1)^(2)-4 (2lambda_(1)-1)` is perfect square for integral roots
`D=lambda_(1)^(2)-8lambda_(1)+4`is perfect square
Let `D= (lambda_(1)-4-k) (lambda_(1)-4+k)=12`
this gives `lambda_(1)-4-k=2`
`(&lambda_(1)-4+k=6)/(lambda_(1)-4 " " 4&k=1)`
`lambda_(1)=8`
`alpha+beta=alphabeta=8-1=7`