The sum of all the roots of the equation  $\left|x^2-8x+15\right|-2x+7=0$ is

Case (i): ${\mathrm{x}}^2-8\mathrm{x}+15>0$

$(x-3)(x-5)>0\Rightarrow x<3,x>5\to \left(1\right)$

G.E $\Rightarrow \left({\mathrm{x}}^2-8\mathrm{x}+15\right)-2\mathrm{x}+7=0$

$\begin{array}{l}{\mathrm{x}}^2-10\mathrm{x}+22=0\\ \mathrm{x}=5+\sqrt{3},5-\sqrt{3}\to \left(2\right)\end{array}$

From (1) & (2) $\Rightarrow \mathrm{x}=5+\sqrt{3}$

Case (ii): $x^2-8x+15<0 \Rightarrow 3<x<5$

Given equation $\Rightarrow -\left({\mathrm{x}}^2-8\mathrm{x}+15\right)-2\mathrm{x}+7=0\to \left(3\right)$

${\mathrm{x}}^2-6\mathrm{x}+8=0\mathrm{x}=2,4\to \left(4\right)$

From (3) & (4)  $\Rightarrow \mathrm{x}=4$

Sum of all roots  $=5+\sqrt{3}+4=9+\sqrt{3}$