In chemical engineering process, water vapor $(H_{2}O)$ is heated to sufficiently high temperature that a

significant portion of the water dissociates, or splits apart, to form oxygen $(O_2)$ and hydrogen $(H_2)$ :

$H_{2}O{H}_2+\frac{1}{2}{O}_2$

If it is assumed that this is the only reaction involved, the mole fraction $x$ of $H_{2}O$ that

dissociates can be represented by $K=\frac{x}{1-x}\sqrt[\phantom{2}]{\frac{2p_t}{2+x}}$ where $K=$ the reaction equilibrium

constant and $p_t=$ the total pressure of the mixture. If $p_t=3$atm and $K=0\mathrm{.}05,$ determine the value of

$x$ that satisfies the equation for $K$ using Newton$-$Raphson Method.