# Linearizing the square root of two binary summations

My question is similar to this one though a bit more complicated. Though my question also includes indices, I am removing them to ease readability.

Let binary variables $$x,y\in\{0,1\}$$, non-negative continuous variable $$z\in\mathbb{R}^+$$ and the parameter $$\lambda\in\mathbb{R}^+$$. Is there a way to linearize the below equality constraint?

$$\displaystyle z=\sqrt{\lambda\left(x+y\right)}$$

Can we benefit from the fact that $$\alpha=x+y$$, where $$\alpha \in \{0,1,2\}$$ and write additional constraints?

For $$j\in\{0,1,2\}$$, introduce binary variable $$w_j$$ to indicate whether $$x+y=j$$, and then impose the following linear constraints: \begin{align} \sum_{j=0}^2 w_j &= 1 \\ \sum_{j=0}^2 j\cdot w_j &= x+y \\ \sqrt{\lambda}\sum_{j=0}^2 \sqrt{j}\cdot w_j &= z \end{align}