If `f:(2,-oo) -> [8, oo)` is a surjective function defined by `f(x) = x^2 - (p-2)x + 3p-2,p in R` then sum of values of p is `m+sqrtn`, where `m, n in N`. Find the value of `n/m`.

Answer

2

Explanation

Case-I : When `(p-2)/(2) le 2`
`implies p le 6`
`f(2) =8`
`implies 4-2(p-2)+3p-2=8`
`implies p=2`
Case-II : When `(p-2)/(2) gt 2 implies p gt 6`
`(-D)/(4a)=8implies -((p-2)^(2)-4(3p-2))/(4)=8`
`implies p^(2)-4p+4-12p+8= -32`
`implies p^(2)-16p+44=0`
`p=(16+-sqrt(256-176))/(2)=(16+-4sqrt(5))/(2)=8+-2sqrt(5)`
` :. p=8+2sqrt(5)`