Why are zeros/roots (real) solutions to an equation of an n-degree polynomial? I can't really put a proper title on this one, but I seem to be missing one crucial point. Why do roots of a function like $f(x) = ax^2 + bx + c$ provide the solutions when $f(x) = 0$. What does that $ y = 0$ mean for the solutions, the intercept at the $x$ axis? Why aren't the solutions at $f(x) = 403045$ or some other arbitrary $n$?
What makes the x-intercept special?

One reason is that it makes solving an equation simple, especially if $f(x)$ is written only as the product of a few terms.  This is because $a\cdot b = 0$ implies either $a = 0$ or $b = 0$. For example, take $f(x) = (x-5)(x+2)(x-2)$.  To find the values of $x$ where $f(x) = 0$ we see that $x$ must be $5$, $-2$, or $2$.  To find the values of $x$ so that $f(x) = 5$, well, we can't conclude anything immediately because having 3 numbers multiply to 5 (or any non-zero number) doesn't tell us anything about those 3 numbers.  This makes 0 special.
