$\text{ Let conditions }{\mathrm{C}}_1\text{ and }{\mathrm{C}}_2\text{ be defined as follows: }{\mathrm{C}}_1:{\mathrm{b}}^2-4\mathrm{ac}\ge 0,{\mathrm{C}}_2:\mathrm{a},-\mathrm{b},\mathrm{c}\text{ are of same sign. The roots of }{\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}=0$

$\mathrm{are} \mathrm{real} \mathrm{and} \mathrm{positive}, \mathrm{if}$

$C_1:b^2-4ac\ge 0$

$C_2:a,-b,c\text{ are of same sign }$

$a{x}^2+bx+c=0\text{ has real roots. }$

$\text{ Then }D\ge 0\text{ i.e., }{C}_1\text{ must be satisfied. }$

$\text{ (i) Let }a,-b,c>0\text{ Then }-\frac{b}{2a}>0$

$\text{ (ii) Let }a,-b,c<0\text{ Then }-\frac{b}{2a}>0$

$\text{ Hence, for roots to be }+\mathrm{ve},C_2\text{ must be satisfied. }$

$\text{ Thus, both }{C}_1,C_2\text{ are satisfied. }$