Why does putting a 1/2 in front of the squared error make the math easier?
A major reason for using MSE is to optimize the parameters of a regression model. From calculus, you know how to find the minimum of a function by taking the derivative. That puts a "2" out in front, which is irritating to keep writing, so it is reasonable to put a "1/2" at the beginning so the derivative doesn't need a constant out front.

We get away with it because the minimum of f(x) and f(x)/2 is achieved at the same value(s) of x.