The smallest positive integral value of a, for which all the roots of x4-ax2+9=0 are real and distinct, is equal to [2026]

Question

The smallest positive integral value of $a$ , for which all the roots of $x^{4} - a x^{2} + 9 = 0$ are real and distinct, is equal to [2026]

Smart Achievers · Accepted Answer

(1)

$x^{4} - a x^{2} + 9 = 0 . . . (1)$

$let x^{2} = t$

$t^{2} - a t + 9 = 0 . . . (2)$

$For roots of equation (1) to be real & distinct, roots of equation (2) must be positive & distinct.$

$(i) D > 0 \Rightarrow a^{2} - 36 > 0 \Rightarrow a \in (- \infty, - 6) \cup (6, \infty)$

$(ii) \frac{- b}{2 a} > 0 \Rightarrow \frac{a}{2} > 0 \Rightarrow a > 0$

$(iii) f (0) > 0 \Rightarrow 9 > 0 \Rightarrow a \in ℝ$

$By (i) \cap (ii) \cap (iii)$

$∴$ $a \in (6, \infty)$

$∴$ $least integral value of a = 7$

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