JEE Main · 2025 · 07 Apr 2025 · Shift 1 · Quadratic Equations and Inequalities

Both roots of $x^2-(p+2)x+(2p+9)=0$ are negative real numbers only when the discriminant is non-negative, the sum of the roots is negative, and their product is positive. The discriminant condition $(p+2{)}^2-4(2p+9)\ge 0$ gives $(p-8)(p+4)\ge 0$, so $p\le -4$ or $p\ge 8$. The sum $p+2<0$ requires $p<-2$, and the product $2p+9>0$ requires $p>-\frac{9}{2}$. Intersecting these leaves $\left(-\frac{9}{2},-4\right]$, so $\alpha =-\frac{9}{2}$ and $\beta =-4$, and $\beta -2\alpha =-4+9=5$.