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Range of Quadratic Expression

If γ, δ are t...

Question

If $γ, δ$ are the roots of $x^{2} - 3 x + a = 0, a ϵ R$ , and $γ < 1 < δ$ then

A

a ϵ (- \infty, 2)

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B

a ϵ (- \infty, \frac{9}{4})

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C

a ϵ (2, \frac{9}{4})

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D

none of these

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Solution

The correct option is A $a ϵ (- \infty, 2)$
Let $γ and δ$ be roots of the equation $x^{2} - 3 x + a = 0$ . The corresponding polynomial of this equation is $f (x) = x^{2} - 3 x + a$

From the figure,
$D > 0 \Rightarrow 9 - 4 a > 0 \Rightarrow a < \frac{9}{4}$
$f (1) < 0 \Rightarrow 1 - 3 + a < 0 \Rightarrow a < 2$
On taking intersection we get,
$a ϵ (- \infty, 2)$

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