1. Tardigrade
  2. Question
  3. Mathematics
  4. Let f(x)=x2+(a+2) x+a2-a+2. Given a, α, β(α<β) be real numbers and α, β are the roots of the equation f ( x )=0. If both roots of the equation f ( x )=0 lies in interval (0, ∞) then range of a, is

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Q. Let $f(x)=x^2+(a+2) x+a^2-a+2$. Given $a, \alpha, \beta(\alpha<\beta)$ be real numbers and $\alpha, \beta$ are the roots of the equation $f ( x )=0$.
If both roots of the equation $f ( x )=0$ lies in interval $(0, \infty)$ then range of $a$, is

Complex Numbers and Quadratic Equations

Solution:

Given, $x^2+(a+2) x+a^2-a+2=0$.
Now, $D >0 \Rightarrow \frac{2}{3}< a <2$ .....(i)
Product of roots $>0 \Rightarrow a ^2- a +2>0$, which is true $\forall a \in R$ .....(ii)
and sum of roots $>0 \Rightarrow-(a+2)>0 \Rightarrow a<-2$.....(iii)
$\therefore $ (i) $\cap($ ii $) \cap($ iii $) \Rightarrow a \in \phi$.