Difference between root, zero and solution.

Solution 1:

Here is my interpretation.

The word solution is used in the following context. Find the solution to $$f(x) = b \tag{$\star$},$$ where $f: A \mapsto B$, i.e., find the set of all $x \in A$ such that $f(x) = b$.

The zeros of the function $f$ is the set of all $a \in A$ such that $f(a) = 0_B$, where $0_B$ is the zero element in $B$. Hence, zeros are used specifically in the context when the $b$ in $(\star)$ is $0_B$. To see it in a slightly different way, the zeros of $g(x)$, where $$g(x) = f(x)-b$$ are the solutions to the equation $f(x) = b$.

The term roots are typically used to describe the zeros of a function, when the function $f(x)$ is of the following form: $f: R \mapsto R$, where $R$ is a ring. I believe, the usage was due to the fact that we talk about finding the square roots, cube roots, etc, which was then extended to polynomials and thereby extended to rings.

Solution 2:

If we're talking about an equation, a root and a solution are synonymous.

If we're talking about a function, a root and a zero are synonymous.