The sum of all the roots of the equation | x ^ 2 - 8x + 15| - 2x +...

Given,

$\left|{x}^{2} - 8x + 15\right| - 2x + 7 = 0$

Now taking Case:$I$ when,

${x}^{2} - 8x + 15 \ge 0 ⇒ ( x - 3 ) ( x - 5 ) \ge 0$

$⇒ x \le 3$ OR $x \ge 5$

\left|{x}^{2} - 8x + 15\right| - 2x + 7 = 0$</p><p>$⇒ {x}^{2} - 8x + 15 - 2x + 7 = 0$</p><p>$⇒ {x}^{2} - 10x + 22 = 0$</p><p>$⇒ x = 5 + \sqrt{3}, 5 - \sqrt{3}\left(\text{rejected}\right)$</p><p>So,$\alpha = 5 + \sqrt{3}$</p><p>Now taking Case:$II$when,</p><p>\left|{x}^{2} - 8x + 15\right| - 2x + 7 = 0$

$⇒ {x}^{2} - 8x + 15 < 0 ⇒ 3 < x < 5$

$⇒ {x}^{2} - 8x + 15 + 2x - 7 = 0$

$⇒ {x}^{2} - 6x + 8 = 0 ⇒ x = 4, 2\left(\text{rejected}\right)$

So, $\gamma = 4$

Hence, $\alpha + \gamma = 9 + \sqrt{3}$