if alpha and beta are the zeros of the polynomial P(x)=x^2-5x+k and Alpha- beta=1, find the value of k.
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let alpha =m
beta =n
now p (m)=m^2-5m+k =0
and p (n )=n^2-5n+k=0
and m-n=1
so frm (1)...k = 5m-m^2
from (2)...k=5n-n^2
so 5m-m^2=5n-n^2
or 5 (m-n)-(m^2-n^2)=5 (m-n)-(m+n)(m-n)=0
or 5 *1-(m+n)*1=0
so m+n=5 and m-n=1
2m=6 or m=3
n=2 means alpha =3 beta =2
beta =n
now p (m)=m^2-5m+k =0
and p (n )=n^2-5n+k=0
and m-n=1
so frm (1)...k = 5m-m^2
from (2)...k=5n-n^2
so 5m-m^2=5n-n^2
or 5 (m-n)-(m^2-n^2)=5 (m-n)-(m+n)(m-n)=0
or 5 *1-(m+n)*1=0
so m+n=5 and m-n=1
2m=6 or m=3
n=2 means alpha =3 beta =2
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