If α and β are negative real roots of the quadratic equation x^2 – (p + 2)x + (2p + 9) = 0 and p ∈ (α, β). Then the value of β^2 - 2α is

Question

If \(\alpha\) and \(\beta\) are negative real roots of the quadratic equation \(x^{2}-(p+2) x+(2 p+9)=0\) and \(p \in(\alpha, \beta).\) Then the value of \(\beta^{2}-2 \alpha\) is

(1) 11

(2) 13

(3) 7

(4) 5 

Answer 1

Correct option is: (2) 13 

\(\alpha \beta>0\)

\(\Rightarrow 2 p+9>0\)

\(\Rightarrow p>\frac{-9}{2}\)

and \(\alpha+\beta<0\)

\(p+2<0 \)

\(\Rightarrow p<-2\)

\(\Rightarrow p \in\left(\frac{-9}{2}-2\right) \)

\(\beta^{2}-2 \alpha=4-2\left(\frac{-9}{2}\right) \)

= 4 + 9 

= 13