Let `a, b, c` be real. If `a x^2+b x+c=0` has two real roots `alpha` and `beta`, where `alpha<<-1` and `beta>>1`, then show that `1+c/a+|b/a|<0`

Let a,b,c be real if ax^(2)+bx+c=0 has two real roots alpha and beta where a lt -2 and beta gt2 , then

Let a,b,c in R and a gt 0 . If the quadratic equation ax^(2) +bx +c=0 has two real roots alpha and beta such that alpha gt -1 and beta gt 1 , then show that 1 + |b/a| + c/a gt 0

ax^(2)+bx+c=0 has real and distinct roots alpha and beta(beta>alpha) .further a>0,b<0,c<0 then

If a,b,c are in A.P and ax^(2)+bx+c=0 has integral roots alpha,beta , then alpha+beta+alpha beta can be equal to (A) -1 (B) 0 (C) 7 (D) -7

Suppose the quadratic polynomial p(x)=ax^(2)+bx+c has positive coefficient a,b,c such that b-a=c-b. If p(x)=0 has integer roots alpha and beta then what could be the possible value of alpha+beta+alpha beta if 0<=alpha+beta+alpha beta<=8

Statement-1: If alpha and beta are real roots of the quadratic equations ax^(2) + bx + c = 0 and -ax^(2) + bx + c = 0 , then (a)/(2) x^(2) + bx + c = 0 has a real root between alpha and beta Statement-2: If f(x) is a real polynomial and x_(1), x_(2) in R such that f(x_(1)) f_(x_(2)) lt 0 , then f(x) = 0 has at leat one real root between x_(1) and x_(2) .

Let equation x^(2)+bx+c=0 has roots alpha and beta such that alpha+beta=3alpha beta Also alpha+5 and beta+5 are roots of the equation x^(2)+ex+f=0 such that (e)/(f)=(1)/(23) , then the value of (b+c) is equal to