Question

Let $a,b,c$ be real numbers such that $a+b+c<0$ and the quadratic equation $a{x}^2+bx+c=0$ has imaginary roots. Then

• A. $a>0,c<0$
• B. $a>0,c>0$
• C. $a<0,c<0$
• D. $a<0,c>0$

Answer 1

The correct option is C $a<0,c<0$

$a{x}^2+bx+c=0$ has imaginary roots
$\Rightarrow$ $f\left(x\right)=a{x}^2+bx+c$ does not intersect $x$-axis for any real $x$. Thus, its graph is either an upward opening parabola lying completely above the $x$-axis or its a downward opening parabola lying completely below the $x$-axis.

Since, $f\left(1\right)=a+b+c<0$.
So, it lies completely below the $x$-axis.
$\Rightarrow a<0$
Also $D<0$
$\Rightarrow b^2-4ac<0$
$\Rightarrow b^2<4ac$
$\Rightarrow a$ and $c$ will have the same sign.