$X_{1}, X_{2}, ..., X_{n}$ is a random sample from $Poisson(\lambda)$ population. I need to show that when sample size n is large, the approximate two-sided (1-$\alpha$)% C.I. is $$ \left[ \bar{x} + \frac{1}{2n} z_{\alpha/2}^2 - \frac{1}{2} \sqrt{\left(2\bar{x}+\frac{1}{n}z_{\alpha/2}^2\right)^2-4\bar{x}^2}, \bar{x} + \frac{1}{2n} z_{\alpha/2}^2 + \frac{1}{2} \sqrt{\left(2\bar{x}+\frac{1}{n}z_{\alpha/2}^2\right)^2-4\bar{x}^2} \right] $$
I got $\frac{\bar{x}-\lambda}{\sqrt{\lambda/n}}\sim N(0,1)$ and $\bar{x}\pm z_{\alpha/2}\sqrt{\bar{x}/n}$ by myself and found from a paper that $\frac{1}{2n}z_{\alpha/2}^2$ is the tail probability for small sample size.
Now I'm confused with $\frac{1}{2}\sqrt{(2\bar{x}+\frac{1}{n}z_{\alpha/2}^2)^2-4\bar{x}^2}$. From the structure of the equation, I guess it is the error? But I'm not sure about it and have no idea how to get it. Can anyone give me some hints or guide for this part please?