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Suppose the quadratic polynomial p(x) =...

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`

A

3

B

5

C

7

D

14

Text Solution

Verified by Experts

The correct Answer is:
C

`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`
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Knowledge Check

  • If alpha and beta are the roots of equation ax^2 + bx + c = 0, then the value of alpha/beta + beta/alpha is

    A
    `(b^2 - 2ac)/(ac)`
    B
    `(b^2 - ac)/(ac)`
    C
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    D
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    A
    `(b^(2)-2ac)/(a^(2)c^(2))`
    B
    `(c^(2)-2ab)/(a^(2)b^(2))`
    C
    `(a^(2)-2bc)/(b^(2)c^(2))`
    D
    none
  • If alpha, beta are the roots of ax^(2)+bx+c=0 , then alpha beta^(2)+ alpha^(2) beta+alpha beta is equal to

    A
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    D
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