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

$$\begin{aligned} &x^2+|2 x-3|-4=0 \\ &|2 x-3|=\left\{\begin{array}{lll} (2 x-3) & \text { if } & \mathrm{x}>\frac{3}{2} \\ -(2 x-3) & \text { if } & \mathrm{x} < \frac{3}{2} \end{array}\right. \end{aligned}$$​x2+∣2x−3∣−4=0∣2x−3∣={(2x−3)−(2x−3)​ if  if ​x>23​x<23​​​ $$\begin{aligned} &\text { for } x>\frac{3}{2}, \quad x^2+2 x-3-4=0 \\ &x^2+2 x-7=0 \\ &x=\frac{-2 \pm \sqrt{4+28}}{2} \\ &=\frac{-2 \pm 4 \sqrt{2}}{2}=-1 \pm 2 \sqrt{2} \\ &\text { Here } x=2 \sqrt{2}-1 \\ &\left\{2 \sqrt{2}-1 < \frac{3}{2}\right\} \\ &\text { for } x < \frac{3}{2} \\ &x^2-2 x+3-4=0 \\ &\Rightarrow x^2-2 x-1=0 \\ &\Rightarrow x=\frac{2 \pm \sqrt{4+4}}{2}=\frac{2 \pm 2 \sqrt{2}}{2}=1 \pm \sqrt{2} \\ &\text { Here } x=1-\sqrt{2} \quad\left\{(1-\sqrt{2}) < \frac{3}{2}\right\} \end{aligned}$$​ for x>23​,x2+2x−3−4=0x2+2x−7=0x=2−2±4+28​​=2−2±42​​=−1±22​ Here x=22​−1{22​−1<23​} for x<23​x2−2x+3−4=0⇒x2−2x−1=0⇒x=22±4+4​​=22±22​​=1±2​ Here x=1−2​{(1−2​)<23​}​ Sum of roots: $(2 \sqrt{2}-1)+(1-\sqrt{2})=\sqrt{2}$(22​−1)+(1−2​)=2​