XGBoost equations (for dummies)
I recall I was struggling for some time deriving the second equation. That constant keeps many of your missing elements. Let's break it down using your $(a-b)^{2}$ notation. We will have $a^{2}$, $b^{2}$, and $2ab$:


$a^{2}$: $y_{i}$ is constant since it is your true labels/values, thus $a^{2}$ i.e. $y_{i}^{2}$  goes to the constant.
$b^{2}$: $y\hat{}_{i}^{t-1}$+$f_{t}^{2}$+$2y\hat{}_{i}^{t-1}f_{t}$


$y\hat{}_{i}^{t-1}$ is constant as it is prediction in a step before $(t-1)$ that we know already, thus goes to the constant term. The other two terms remain as it is.


$2ab$: $2y_{i}(y\hat{}_{i}^{t-1}+f_{t}) = -2y_{i}y\hat{}_{i}^{t-1}-2y_{i}f_{t}$. Here also the first term is constant. Only the second term remains.


The rest should be pretty straightforward, just add things that are left and clean it up and you see the second equation comes out beautifully.