Number of integral values of $x$ satisfying $| x^{2} - 5 x + 4 | = 2$ is

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Solution

$| x^{2} - 5 x + 4 | = 2$
If $x^{2} - 5 x + 4 < 0$ i.e., $x \in (1, 4)$
then $- (x^{2} - 5 x + 4) = 2$
$\Rightarrow x^{2} - 5 x + 6 = 0$
$\Rightarrow (x - 3) (x - 2) = 0$
$\Rightarrow x = 3, 2$

If $x^{2} - 5 x + 4 \geq 0$ i.e., $x \in (- \infty, 1] \cup [4, \infty)$
then $x^{2} - 5 x + 4 = 2$
$\Rightarrow x^{2} - 5 x + 2 = 0 \Rightarrow x = \frac{5 \pm \sqrt{25 - 4 \times 2}}{2} \Rightarrow x = \frac{5 \pm \sqrt{17}}{2}$

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