Question

Find the condition that $a{x}^2+bx+c=0$ may have
(i) both roots positive (ii) both negative roots
(iii) roots are equal in magnitude but opposite in sign.

Answer 1

Given

$a{x}^3+bx+c=0$

$C-I$ (If $a>0$)

i) $\alpha +\beta =\frac{-b}{a},\alpha \beta >0$

$\frac{-b}{a}>0\frac{c}{a}>0$

$\therefore \frac{b}{a}<0c>0$

$\therefore b<0$

$C-II$ (If $a<0$)

$b>0,c<0$

ii) $C-I$ $(a>0)$

$\alpha +\beta <0$

$\frac{-b}{a}<0\alpha \beta >0$

$b>0,c>0$

$C-II$ $(a<0)$

$\frac{-b}{a}<0,\alpha \beta >0$

$\therefore b<0c<0$

iii) $\alpha +\beta =0$

$\therefore b=0$