Bayesian parameter estimation for simple linear regression

Question

I'm reading William Bolstad's introductory book to Bayesian statistics and I don't quite understand how did we jump from $e^{\frac{1}{2 \sigma ^2} (SS_y - 2 \beta SS_{xy} + \beta ^2 SS_x)}$ to $e^{\frac{1}{2 \sigma ^2 / SS_x} (\beta - \frac{SS_{xy}}{SS_x})^2}$, the factorisation part was easy for me to understand $e^{\frac{1}{2 \sigma ^2 / SS_x} (\beta ^2 - 2 \beta \frac{SS_{xy}}{SS_x} + \frac{SS_y}{SS_x})}$, but the two following parts is where I got stuck. Can anyone show, step by step, how did we get there starting from $e^{\frac{1}{2 \sigma ^2 / SS_x} (\beta ^2 - 2 \beta \frac{SS_{xy}}{SS_x} + \frac{SS_y}{SS_x})}$?

Answer 1

Starting from $e^{\frac{1}{2 \sigma ^2 / SS_x} (\beta ^2 - 2 \beta \frac{SS_{xy}}{SS_x} + \frac{SS_y}{SS_x})}$, they have expressed this as follows:- $$\begin{align}e^{\frac{1}{2 \sigma ^2 / SS_x} (\beta ^2 - 2 \beta \frac{SS_{xy}}{SS_x} + \frac{SS_y}{SS_x}+\left(\frac{SS_{xy}}{SS_x}\right)^2-\left(\frac{SS_{xy}}{SS_x}\right)^2)}=e^{\frac{1}{2 \sigma ^2 / SS_x} (\beta - \frac{SS_{xy}}{SS_x})^2 }\color{red}{e^{\frac{1}{2 \sigma ^2 / SS_x} (\frac{SS_y}{SS_x}-\left(\frac{SS_{xy}}{SS_x}\right)^2) }}\end{align}$$ The part that doesn't depend on any parameter is highlighted in red, and has been absorbed into the constant of proportionality.