The sum of the solutions of the equation | x - 2| + x ( x - 4) + 2...

Given,

$\left|\sqrt{x} - 2\right| + \sqrt{x}\left(\sqrt{x} - 4\right) + 2 = 0$
$⇒ \left|\sqrt{x} - 2\right| + x - 4\sqrt{x} + 2 = 0$
$⇒ \left|\sqrt{x} - 2\right| + {\left(\sqrt{x}\right)}^{2} - 4\sqrt{x} + 4 = 2$ (Adding $2$ on both sides)
$⇒ {\left(\sqrt{x} - 2\right)}^{2} + \left|\sqrt{x} - 2\right| = 2$
$⇒ {t}^{2} + t - 2 = 0$; \left|\sqrt{x} - 2\right| = t; \left(t > 0\right)$<br />$⇒ t = 1$or$t = - 2$; Rejected because$t > 0$<br />$⇒ \left|\sqrt{x} - 2\right| = 1$<br />$⇒ \sqrt{x} = 3 or $\sqrt{x} = 1$
$⇒ x = 9 or x = 1$
Hence, the required sum $= 10$.