Question

What is the condition that the equation $a{x}^2+bx+c=0$, where $a\ne 0$, has both the roots positive?

• A. $a,b$ and $c$ are of same sign
• B. $a$ and $b$ are of same sign
• C. $b$ and $c$ have the same sign opposite to that of $a$
• D. $a$ and $c$ have the same sign opposite to that of $b$

Answer 1

The correct option is D $a$ and $c$ have the same sign opposite to that of $b$

$\alpha ,\beta$ the roots of $a{x}^2+bx+c=0$ are positive.
$\alpha +\beta =-\frac{b}{a}$ .... (i)
$\alpha \beta =\frac{c}{a}$ ..... (ii)
Clearly $\frac{c}{a}$ is $+ve$.

So, $c$ and $a$ must have the same sign but opposite to that of '$b$' since $\alpha +\beta$ is $-ve$.