S(\alpha, \beta) = 
\left( \sum_i^n x_i^2 \right) \alpha^2 + n\beta^2
+ 2 \left( \sum_i^n x_i \right)\alpha \beta
- 2 \left( \sum_i^n x_i y_i \right)\alpha 
- 2 \left( \sum_i^n y_i \right)\beta
+ \sum_i^n y_i^2