Question

Let `a ,b ,c` be real. If `a x^2+b x+c=0` has two real roots `alphaa n dbeta,w h e r ealpha<<-1a n dbeta>>1` , then show that `1+c/a+|b/a|<0`

Answer 1

Let `f(x)= x^(2)+ (b)/(a) x + (c)/(a)`.
According to the questions, we have the following graph.

From the graph
`" "f(-1) lt 0 and f(1) lt 0`
`rArr" "1 + (c)/(a)- (b)/(a)lt 0 and 1+ (c)/(a)+ (b)/(a) lt 0`
`rArr" "1+ (c)/(a)+ |(b)/(a)| lt 0`