The sum of all the roots of the equation x 2 - 8 x + 15 - 2 x + 7 = 0 is

Given,

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

Now taking Case:$\mathrm{I}$ when,

$x^2-8x+15\ge 0\Rightarrow (x-3)(x-5)\ge 0$

$\Rightarrow x\le 3$ OR $x\ge 5$

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

$\Rightarrow x^2-8x+15-2x+7=0$

$\Rightarrow x^2-10x+22=0$

$\Rightarrow x=5+\sqrt{3}, 5-\sqrt{3} \left(\text{rejected}\right)$ 

So, $\alpha =5+\sqrt{3}$

Now taking Case:$\mathrm{II}$ when,

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

$\Rightarrow x^2-8x+15<0\Rightarrow 3<x<5$

$\Rightarrow x^2-8x+15+2x-7=0$

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

So, $\gamma =4$

Hence, $\alpha +\gamma =9+\sqrt{3}$