Question

If $a,b,c\in R$ and the quadratic equation $a{x}^2+bx+c=0$ has no real roots, then

• A. $(a+b+c)c>0$
• B. $c<0$
• C. $a+b+c>0$
• D. $a+b+c<0$

Answer 1

The correct option is A $(a+b+c)c>0$

Let $f\left(x\right)=a{x}^2+bx+c$
For $a>0$,
$f\left(x\right)>0$, for all $x$
Hence,
$f\left(1\right)=a+b+c>0$
$f\left(0\right)=c>0$
For $a<0$ , $f\left(x\right)<0$
$f\left(1\right)=a+b+c<0$
$f\left(0\right)=c<0$
In both the cases we have $c(a+b+c)>0$