Both roots are positive
D > 0, S > 0, P > 0
Sum of the roots, S = 0 ⇒ (a-1) > 0 ⇒ a < 1
Product of roots, P > 0 ⇒ a > 0
Step-by-Step Solution
Key Concept: To ensure both roots of a quadratic are positive, check three conditions simultaneously: non-negative discriminant, positive sum of roots, and positive product of roots.
For a quadratic with both roots positive, we need $D > 0$, $S > 0$, and $P > 0$. Discriminant: $D = 0 \Rightarrow (a-1)^2 - 4a > 0 \Rightarrow a^2 + 1 - 6a > 0$. Sum of roots: $S > 0 \Rightarrow -(a-1) > 0 \Rightarrow a < 1$. Product of roots: $P > 0 \Rightarrow a > 0$. Combining all conditions gives $0 < a < 3 - 2\sqrt{2}$.
Correct Answer: 3