InterviewSolution
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InterviewSolution
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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. 3B. 5C. 7D. 14 |
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Answer» Correct Answer - 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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