Question

The quadratic equation $a{x}^2+bx+c=0$ will always have imaginary root in which of the following cases?

• A. $a<-1,0<c<1,b>0$
• B. $a<-1,1<c<0,0<b<1$
• C. $a<-1,c<0,b>1$
• D. $a<-1,c<-1,1<b<2$

Answer 1

Answer: (D)

$a<-1,c<-1,1<b<2$

$f\left(x\right)=a{x}^2+bx+c=0$

Roots are imaginary therefore discriminant is less than zero

$b^2-4ac<0$

If $a>0$ so upward parabola and $f\left(x\right)>0$ $\forall x\in R$ - 1

If $a<0$ so downward parabola and $f\left(x\right)<0$ $\forall x\in R$ - 2

As per given options a<-1, condition 2 will be satisfied

$b^2-4ac<0$ is always positive so ac is also positive as a is negative, $C$ is also negative If $c<-1,1<b<2$

$a<-1,c<-1,1<b<2$