Question

If $a,b,c\in R$ and $a\ne 0,c<0$, and if the quadratic equation $a{x}^2+bx+c=0$ has imaginary roots, then $a+b+c$ is

• A. Can't say
• B. Negative
• C. Zero

Answer 1

The correct option is B

Negative

Let $y=a{x}^2+bx+c$

It's given that $a{x}^2+bx+c=0$ has only imaginary roots. That means the graph of $y=a{x}^2+bx+c$ does not touch the $x$-axis.

It either completely lies above the $x$-axis or lies below the $x$-axis. We are given $c<0$.

$c$ is the value of $y$ when $x=0$.

That means the graph lies completely below $x$-axis or the value of $y=a{x}^2+bx+c$ is less than zero for any value of $x$.

It is less than zero for $x=1$ also. Value of $y$ when $x=1$ is $a+b+c$.

That means $a+b+c<0$.