If the equation $x^{2} + 2 | a | x + 4 = 0$ has integral roots then the minimum value of a is

$- \frac{5}{2}$

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Solution

The correct option is A $- \frac{5}{2}$
Given equation $x^{2} + 2 | a | x + 4 = 0$
Since, the condition for integral roots is $b^{2} - 4 a c = k^{2}$
$\Rightarrow (2 | a |)^{2} - 4 (1) (4) = k^{2}$
$\Rightarrow 4 a^{2} - 16 = k^{2}$
$\Rightarrow 4 a^{2} = k^{2} + 16$
$\Rightarrow a^{2} = \frac{k^{2} + 16}{4}$
$\Rightarrow a = \pm \sqrt{\frac{k^{2} + 16}{4}}$
Take $k^{2} = 9$
$\Rightarrow a = \pm \sqrt{\frac{9 + 16}{4}}$
$∴ a = \pm \frac{5}{2}$

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