Mathematic explanation needed for a univariate classification method based on solving a quadratic equation
Actually, it is very straightforward to derive. 

Hint 1: where would you most likely find mean and variance used? Gaussian!

Hint 2: how would you do classification using Gaussians? Simple, you fit a Gaussian to each group and compare the PDFs of new data $x$. Even simpler, you fit a Gaussian to each group and calculate the Mahalanobis distance of $x$ to each group.



A threshold is called a threshold because at this value you cannot tell which class it is in, aka, the Mahalanobis distances to two groups equal:

$$\frac{(x-\mu_A)^2}{\sigma_A^2}=\frac{(x-\mu_B)^2}{\sigma_B^2}$$

Expanding this, you will get the quadratic formula.