Question

If the roots of x^2 -px +q=0 are two consequtive integer then prove that p^2=4q +1

Answer 1

let the first root b a thn other root be a+1
a+a+1=p
2a+1=p
a=(p-1)/2

a (a+1)=q
a^2+a=q
{(p-1)/2}^2+(p-1)/2=q
(p^2+1-2p+2p-2)/4=q [Taking LCM]
p^2-1=4q
p^2=4q+1
Hence Proved