Question

If both the roots of the quadratic equation $a{x}^2+bx+c=0$ are negative, then

• A. $a,b,c$ can be any positive rational numbers
• B. $a,b,c$ can be any integers
• C. $a,b,c$ can be any negative integers
• D. $a,b,c$ can be any real numbers

Answer 1

Answer: (A), (C)

$a,b,c$ can be any negative integers

As both the roots are negative, so
$\alpha +\beta <0\Rightarrow -\frac{b}{a}<0\Rightarrow \frac{b}{a}>0\cdots \left(1\right)\alpha \times \beta >0\Rightarrow \frac{c}{a}>0\cdots \left(2\right)$

When $a>0$, using equation $\left(1\right)$ and $\left(2\right)$, we get
$b>0,c>0$
Similarly, when $a<0$, using equation $\left(1\right)$ and $\left(2\right)$, we get
$b<0,c<0$

Hence, $a,b,c$ has to be of same sign.