Find the number of integral values of $a$ so that the inequation $x^{2} - 2 (a + 1) x + 3 (a - 3) (a + 1) < 0$ is satisfied by atleast one $x \in R^{+}$ .

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Solution

$x^{2} - 2 (a + 1) x + 3 (a - 3) (a + 1) < 0$
This inequality holds when $(a + 1)^{2} - 3 (a - 3) (a + 1) < 0 ⟹ (a + 1) (a - 4) > 0$
$a \in (- \infty, - 1) \cup (4, \infty)$
Infinite no of integral values are there

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