Roots Negative? Here’s the Fastest Way to Solve This p-Parameter Question ⚡
❓ Question Let the set of all values of p ∈ R p \in \mathbb{R} for which both the roots of the equation x 2 − ( p + 2 ) x + ( 2 p + 9 ) = 0 are negative real numbers be the interval ( α , β ] (\alpha, \beta] . Then the value of β − 2 α is equal to ? 🖼️ Question Image ✍️ Short Solution We have a quadratic: x 2 − ( p + 2 ) x + ( 2 p + 9 ) = 0 Compare with x 2 − S x + P = 0 x^2 - Sx + P = 0 , where Sum of roots = S = p + 2 = S = p+2 Product of roots = P = 2 p + 9 = P = 2p+9 For both roots to be negative real numbers , we need: Real roots → Discriminant Δ ≥ 0 Both roots negative → Sum of roots < 0 <0 Product of roots > 0 >0 🔹 Step 1 — Conditions from sum and product Sum < 0: p + 2 < 0 ⇒ p < − 2 Product > 0: 2 p + 9 > 0 ⇒ p > − 9 2 Together: − 9 2 < p < − 2 🔹 Step 2 — Discriminant condition Δ = ( p + 2 ) 2 − 4 ( 2 p + 9 ) ≥ 0 Compute: Δ = ( p 2 + 4 p + 4 ) − ( 8 p + 36 ) = p 2 − 4 p − 32 So...