Hi
Maybe I can add something
The product of the roots (real or complex) of equation is
Therefore has 2 real roots with one positive and one negative if both conditions are realised :
(i)
(ii)
Condition (ii) implies that a and c have different sign ; then their product is negative
It means that condition (ii) implies condition (i) and therefore only condition (ii) is sufficient
In your specific case has 2 real roots with one positive and one negative if then