Why are equations written by equating something to zero? A linear equation is $$ ax + b = 0 ; \,\, \,\, a\neq 0 $$
A quadratic equation is $$ax^2 + bx + c = 0 ; \,\, a\neq 0 $$
And so on...
Why are all these equations written as $\dots = 0 $? Why do mathematicians do it this way?

It's simply a way of putting an equation into a standard form.  You can always add and subtract the same quantities from both sides so that one of the sides becomes zero without changing the solution(s) of the equation.
