If one root of the equation ax2+bx+c=0  is n  times the other, then

The given equation is ax2+bx+c=0  ….. ( 1 ) By the given condition: the roots of ( 1 ) are α,nα  then sum of the roots : α+nα=−ba   ⇒  α(n+1)=  −ba      ..(2) And  product of the roots : α (nα)  =  ca  ⇒nα2=  ca   .. (3)Form ( 2 ) :  α  =  −ba(n+1) Substituting this value of α in (3),  we get n[−ba(n+1)]2  =  ca ⇒  nb2=ca  (n+1)2  ⇒    nb2=ca  (n2+1+2n) ⇒    nb2−2nac=  (n2+1)  ac. ⇒  n(b2−2ac)  =  (n2+1)ac.