Question

Given that $a{x}^2+bx+c=0$ has no real roots and $a+b+c<0$, then

• A. $c=0$
• B. $c>0$
• C. $c<0$
• D. $c\ne 0$

Answer 1

Answer: (C)

Since the equation $a{x}^2+bx+c$ = 0 has no real roots.
$\therefore$ the curve $y=a{x}^2+bx+c$ does not meet $x$-axis.
$\therefore f(x)=a{x}^2+bx+c$ has the same sign for all values of $x$.
But $a+b+c<0$ and $f(1)=a+b+c<0$
$\therefore f(x)<0$ for all $x$,
$\therefore f(0)<0$
$\Rightarrow c<0$