1. Tardigrade
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  3. Mathematics
  4. If α+(1/α), β+(1/β) are zeros of f ( x )= x 2-5 x - a, where α, β ∈(0, ∞) for all x ∈ R, then the complete set of values of a is

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Q. If $\alpha+\frac{1}{\alpha}, \beta+\frac{1}{\beta}$ are zeros of $f ( x )= x ^2-5 x - a$, where $\alpha, \beta \in(0, \infty)$ for all $x \in R$, then the complete set of values of a is

Complex Numbers and Quadratic Equations

Solution:

Let $f ( x )= x ^2-5 x - a$
$\therefore$ We must have
$D \geq 0 \Rightarrow a \geq \frac{-25}{4} $ .....(1)
$f (2) \geq 0 \Rightarrow a \leq-6 $ .....(2)
$\text { and } \quad \frac{- B }{2 A } \geq 2 \Rightarrow \frac{5}{2} \geq 2
$ .....(3)
must be satisfied simultaneously
$\therefore(1) \cap(2) \cap(3) \Rightarrow \alpha \in\frac{-25}{4},-6$