The sum of the roots of the equation, x^2+|2 x-3|-4=0 , is:

\begin{aligned} &x^2+|2 x-3|-4=0 \\ &|2 x-3|=\left\{\begin{array}{lll} (2 x-3) & \text{ if } & \mathrm{x}>\frac{3}{2} \\ -(2 x-3) & \text{ if } & \mathrm{x} < \frac{3}{2} \end{array}\right. \end{aligned}$$\begin{aligned} &\text{ for } x>\frac{3}{2}, \quad x^2+2 x-3-4=0 \\ &x^2+2 x-7=0 \\ &x=\frac{-2 \pm \sqrt{4+28}}{2} \\ &=\frac{-2 \pm 4 \sqrt{2}}{2}=-1 \pm 2 \sqrt{2} \\ &\text{ Here } x=2 \sqrt{2}-1 \\ &\left\{2 \sqrt{2}-1 < \frac{3}{2}\right\} \\ &\text{ for } x < \frac{3}{2} \\ &x^2-2 x+3-4=0 \\ &\Rightarrow x^2-2 x-1=0 \\ &\Rightarrow x=\frac{2 \pm \sqrt{4+4}}{2}=\frac{2 \pm 2 \sqrt{2}}{2}=1 \pm \sqrt{2} \\ &\text{ Here } x=1-\sqrt{2} \quad\left\{(1-\sqrt{2}) < \frac{3}{2}\right\} \end{aligned} $$Sum of roots:$(2 \sqrt{2}-1)+(1-\sqrt{2})=\sqrt{2}$