For the question: Find all values of p so that x^2 - 2px + p - 2 = 0 has one positive root and a negative root.
What is the condition of the discriminant for a positive and a negative root?
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 :
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